From 8d4fe2bb1a7c678cdc2ae09bf16fc17f16a27ae4 Mon Sep 17 00:00:00 2001
From: Vincent Rouvreau <vincent.rouvreau@inria.fr>
Date: Tue, 24 Sep 2024 10:00:15 +0200
Subject: [PATCH 1/6] feature requires python 3.8

---
 .github/next_release.md         |  3 ++-
 src/python/doc/installation.rst | 42 ++++++++++++++++-----------------
 src/python/setup.py.in          |  2 +-
 3 files changed, 24 insertions(+), 23 deletions(-)

diff --git a/.github/next_release.md b/.github/next_release.md
index abd2ebd1bd..573d8ef940 100644
--- a/.github/next_release.md
+++ b/.github/next_release.md
@@ -20,6 +20,7 @@ Below is a list of changes:
 
 - Installation
      - CMake &ge; 3.15 is now required (was &ge; 3.8).
+     - Python &ge; 3.8 is now required (was &ge; 3.5), because of `importlib.metadata`.
 
 - Miscellaneous
      - The [list of bugs that were solved](https://github.com/GUDHI/gudhi-devel/issues?q=label%3A3.11.0+is%3Aclosed) is available on GitHub.
@@ -29,7 +30,7 @@ However, there are still GPL dependencies for many modules. We invite you to che
 
 We kindly ask users to cite the GUDHI library as appropriately as possible in their papers, and to mention the use of the GUDHI library on the web pages of their projects using GUDHI and provide us with links to these web pages.
 
-We provide [bibtex entries](https://gudhi.inria.fr/doc/latest/_citation.html) for the modules of the User and Reference Manual, as well as for publications directly related to the GUDHI library. 
+We provide [bibtex entries](https://gudhi.inria.fr/doc/latest/_citation.html) for the modules of the User and Reference Manual, as well as for publications directly related to the GUDHI library.
 
 Feel free to [contact us](https://gudhi.inria.fr/contact/) in case you have any questions or remarks.
 
diff --git a/src/python/doc/installation.rst b/src/python/doc/installation.rst
index 639a89d652..c1f7bd2c44 100644
--- a/src/python/doc/installation.rst
+++ b/src/python/doc/installation.rst
@@ -41,7 +41,7 @@ there.
 
 The library uses c++17 and requires `Boost <https://www.boost.org/>`_ :math:`\geq` 1.71.0,
 `CMake <https://www.cmake.org/>`_ :math:`\geq` 3.15,
-Python :math:`\geq` 3.5, `NumPy <http://numpy.org>`_ :math:`\geq` 1.15.0, `Cython <https://www.cython.org/>`_
+Python :math:`\geq` 3.8, `NumPy <http://numpy.org>`_ :math:`\geq` 1.15.0, `Cython <https://www.cython.org/>`_
 :math:`\geq` 0.27 and `pybind11 <https://github.com/pybind/pybind11>`_ to compile the GUDHI Python module.
 It is a multi-platform library and compiles on Linux, Mac OSX and Visual Studio 2017 or later.
 
@@ -145,10 +145,10 @@ You shall have something like:
 
 .. code-block:: none
 
-    Pybind11 version 2.8.1 
+    Pybind11 version 2.8.1
     Python version 3.7.12
-    Cython version 0.29.25 
-    Numpy version 1.21.4 
+    Cython version 0.29.25
+    Numpy version 1.21.4
     Boost version 1.77.0
     + Installed modules are: off_utils;simplex_tree;rips_complex;cubical_complex;periodic_cubical_complex;
         persistence_graphical_tools;reader_utils;witness_complex;strong_witness_complex;
@@ -156,7 +156,7 @@ You shall have something like:
         euclidean_strong_witness_complex;
 
 Here, you can see that the modules that need CGAL are missing, because CGAL is not installed.
-:code:`persistence_graphical_tools` is installed, but 
+:code:`persistence_graphical_tools` is installed, but
 `its functions <https://gudhi.inria.fr/python/latest/persistence_graphical_tools_ref.html>`_ will produce an error as
 matplotlib is not available.
 Unitary tests cannot be run as pytest is missing.
@@ -165,23 +165,23 @@ A complete configuration would be :
 
 .. code-block:: none
 
-    Pybind11 version 2.8.1 
+    Pybind11 version 2.8.1
     Python version 3.9.7
-    Cython version 0.29.24 
-    Pytest version 6.2.5 
-    Matplotlib version 3.5.0 
-    Numpy version 1.21.4 
-    Scipy version 1.7.3 
-    Scikit-learn version 1.0.1 
-    POT version 0.8.0 
+    Cython version 0.29.24
+    Pytest version 6.2.5
+    Matplotlib version 3.5.0
+    Numpy version 1.21.4
+    Scipy version 1.7.3
+    Scikit-learn version 1.0.1
+    POT version 0.8.0
     HNSWlib found
     PyKeOps version [pyKeOps]: 2.1
-    EagerPy version 0.30.0 
-    TensorFlow version 2.7.0 
-    Sphinx version 4.3.0 
-    Sphinx-paramlinks version 0.5.2 
-    pydata_sphinx_theme version 0.13.1 
-    NetworkX version 3.0 
+    EagerPy version 0.30.0
+    TensorFlow version 2.7.0
+    Sphinx version 4.3.0
+    Sphinx-paramlinks version 0.5.2
+    pydata_sphinx_theme version 0.13.1
+    NetworkX version 3.0
     Eigen3 version 3.4.0
     Boost version 1.74.0
     CGAL version 5.3
@@ -194,7 +194,7 @@ A complete configuration would be :
     + Installed modules are: bottleneck;off_utils;simplex_tree;rips_complex;cubical_complex;periodic_cubical_complex;
         persistence_graphical_tools;reader_utils;witness_complex;strong_witness_complex;nerve_gic;subsampling;
         tangential_complex;alpha_complex;euclidean_witness_complex;euclidean_strong_witness_complex;
-    + Missing modules are: 
+    + Missing modules are:
 
 
 Documentation
@@ -422,7 +422,7 @@ TensorFlow
 requires `TensorFlow <https://www.tensorflow.org/>`_.
 The :doc:`cubical complex </cubical_complex_tflow_itf_ref>`, :doc:`simplex tree </ls_simplex_tree_tflow_itf_ref>`
 and :doc:`Rips complex </rips_complex_tflow_itf_ref>` modules require `TensorFlow`_
-for incorporating them in neural nets. 
+for incorporating them in neural nets.
 
 `TensorFlow`_ is also used in some automatic differentiation tests.
 
diff --git a/src/python/setup.py.in b/src/python/setup.py.in
index 3bc8d0805e..715ecdef77 100644
--- a/src/python/setup.py.in
+++ b/src/python/setup.py.in
@@ -73,7 +73,7 @@ setup(
     long_description_content_type='text/x-rst',
     long_description='@GUDHI_LONG_DESCRIPTION@',
     ext_modules = ext_modules,
-    python_requires='>=3.5.0',
+    python_requires='>=3.8.0',
     install_requires = ['numpy >= 1.15.0',],
     package_data={"": ["*.dll"], },
 )

From 7b6d033c934ef927bac92e2753d7e904dbdbfa5f Mon Sep 17 00:00:00 2001
From: Vincent Rouvreau <vincent.rouvreau@inria.fr>
Date: Tue, 24 Sep 2024 11:53:59 +0200
Subject: [PATCH 2/6] Use importlib.metadata to guess version numbers of
 dependencies

---
 .../modules/GUDHI_third_party_libraries.cmake | 30 +++------------
 src/python/CMakeLists.txt                     | 34 ++++++++---------
 src/python/doc/installation.rst               | 38 ++++++++++---------
 3 files changed, 43 insertions(+), 59 deletions(-)

diff --git a/src/cmake/modules/GUDHI_third_party_libraries.cmake b/src/cmake/modules/GUDHI_third_party_libraries.cmake
index 7c03103cba..18149ebcc2 100644
--- a/src/cmake/modules/GUDHI_third_party_libraries.cmake
+++ b/src/cmake/modules/GUDHI_third_party_libraries.cmake
@@ -145,8 +145,9 @@ if (WITH_GUDHI_PYTHON)
   # returns ${PYTHON_MODULE_NAME_UP}_VERSION and ${PYTHON_MODULE_NAME_UP}_FOUND
   function( find_python_module PYTHON_MODULE_NAME )
     string(TOUPPER ${PYTHON_MODULE_NAME} PYTHON_MODULE_NAME_UP)
+    # Modify tracebacklimit as the exception is quite verbose when module is not found
     execute_process(
-            COMMAND ${Python_EXECUTABLE}  -c "import ${PYTHON_MODULE_NAME}; print(${PYTHON_MODULE_NAME}.__version__)"
+            COMMAND ${Python_EXECUTABLE}  -c "import sys; sys.tracebacklimit = 0; from importlib.metadata import version; print(version('${PYTHON_MODULE_NAME}'))"
             RESULT_VARIABLE PYTHON_MODULE_RESULT
             OUTPUT_VARIABLE PYTHON_MODULE_VERSION
             ERROR_VARIABLE PYTHON_MODULE_ERROR)
@@ -167,25 +168,6 @@ if (WITH_GUDHI_PYTHON)
     endif()
   endfunction( find_python_module )
 
-  # For modules that do not define module.__version__
-  function( find_python_module_no_version PYTHON_MODULE_NAME )
-    string(TOUPPER ${PYTHON_MODULE_NAME} PYTHON_MODULE_NAME_UP)
-    execute_process(
-            COMMAND ${Python_EXECUTABLE}  -c "import ${PYTHON_MODULE_NAME}"
-            RESULT_VARIABLE PYTHON_MODULE_RESULT
-            ERROR_VARIABLE PYTHON_MODULE_ERROR)
-    if(PYTHON_MODULE_RESULT EQUAL 0)
-      # Remove carriage return
-      message ("++ Python module ${PYTHON_MODULE_NAME} found")
-      set(${PYTHON_MODULE_NAME_UP}_FOUND TRUE PARENT_SCOPE)
-    else()
-      message ("PYTHON_MODULE_NAME = ${PYTHON_MODULE_NAME}
-       - PYTHON_MODULE_RESULT = ${PYTHON_MODULE_RESULT}
-       - PYTHON_MODULE_ERROR = ${PYTHON_MODULE_ERROR}")
-      set(${PYTHON_MODULE_NAME_UP}_FOUND FALSE PARENT_SCOPE)
-    endif()
-  endfunction( find_python_module_no_version )
-
   if( TARGET Python::Interpreter )
     find_python_module("cython")
     find_python_module("pytest")
@@ -193,17 +175,17 @@ if (WITH_GUDHI_PYTHON)
     find_python_module("numpy")
     find_python_module("scipy")
     find_python_module("sphinx")
-    find_python_module("sklearn")
-    find_python_module("ot")
+    find_python_module("scikit-learn")
+    find_python_module("POT")
     find_python_module("pybind11")
     find_python_module("torch")
     find_python_module("pykeops")
     find_python_module("eagerpy")
-    find_python_module_no_version("hnswlib")
+    find_python_module("hnswlib")
     find_python_module("tensorflow")
     find_python_module("sphinx_paramlinks")
     find_python_module("pydata_sphinx_theme")
-    find_python_module_no_version("sphinxcontrib.bibtex")
+    find_python_module("sphinxcontrib.bibtex")
     find_python_module("networkx")
   endif()
 
diff --git a/src/python/CMakeLists.txt b/src/python/CMakeLists.txt
index 5b2fad86aa..e68e8f2c64 100644
--- a/src/python/CMakeLists.txt
+++ b/src/python/CMakeLists.txt
@@ -131,19 +131,19 @@ if(SCIPY_FOUND)
 else()
   disable_python_documentation("scipy")
 endif()
-if(SKLEARN_FOUND)
-  add_gudhi_debug_info("Scikit-learn version ${SKLEARN_VERSION}")
+if(SCIKIT-LEARN_FOUND)
+  add_gudhi_debug_info("Scikit-learn version ${SCIKIT-LEARN_VERSION}")
 else()
-  disable_python_documentation("sklearn")
+  disable_python_documentation("scikit-learn")
 endif()
-if(OT_FOUND)
-  add_gudhi_debug_info("POT version ${OT_VERSION}")
+if(POT_FOUND)
+  add_gudhi_debug_info("POT version ${POT_VERSION}")
 else()
   disable_python_documentation("ot")
 endif()
 if(HNSWLIB_FOUND)
   # Does not have a version number...
-  add_gudhi_debug_info("HNSWlib found")
+  add_gudhi_debug_info("HNSWlib version ${HNSWLIB_VERSION}")
 endif()
 if(TORCH_FOUND)
   add_gudhi_debug_info("PyTorch version ${TORCH_VERSION}")
@@ -176,7 +176,7 @@ else()
 endif()
 if(SPHINXCONTRIB.BIBTEX_FOUND)
   # Does not have a version number...
-  add_gudhi_debug_info("sphinxcontrib-bibtex found")
+  add_gudhi_debug_info("sphinxcontrib-bibtex version ${SPHINXCONTRIB.BIBTEX_VERSION}")
 else()
   disable_python_documentation("sphinxcontrib-bibtex")
 endif()
@@ -461,7 +461,7 @@ add_test(NAME random_cubical_complex_persistence_example_py_test
 add_gudhi_py_test(test_cubical_complex)
 
 # Datasets are fetched for these tests
-if(SKLEARN_FOUND AND WITH_GUDHI_REMOTE_TEST)
+if(SCIKIT-LEARN_FOUND AND WITH_GUDHI_REMOTE_TEST)
   add_gudhi_py_test(test_sklearn_cubical_persistence)
 
   add_test(NAME cubical_complex_sklearn_itf_py_test
@@ -590,7 +590,7 @@ add_test(NAME rips_complex_from_points_example_py_test
 
 add_gudhi_py_test(test_rips_complex)
 
-if(SKLEARN_FOUND)
+if(SCIKIT-LEARN_FOUND)
   # test_sklearn_rips_persistence is using gudhi.datasets.generators
   if (TARGET CGAL::CGAL AND TARGET Eigen3::Eigen)
     add_gudhi_py_test(test_sklearn_rips_persistence)
@@ -636,7 +636,7 @@ add_gudhi_py_test(test_reader_utils)
 add_gudhi_py_test(test_off)
 
 # Wasserstein
-if(OT_FOUND)
+if(POT_FOUND)
   # EagerPy dependency because of enable_autodiff=True
   if(EAGERPY_FOUND)
     add_gudhi_py_test(test_wasserstein_distance)
@@ -648,7 +648,7 @@ if(OT_FOUND)
 endif()
 
 # Representations
-if(SKLEARN_FOUND AND MATPLOTLIB_FOUND AND OT_FOUND AND TARGET CGAL::CGAL)
+if(SCIKIT-LEARN_FOUND AND MATPLOTLIB_FOUND AND POT_FOUND AND TARGET CGAL::CGAL)
   add_gudhi_py_test(test_representations)
 endif()
 
@@ -658,17 +658,17 @@ if(TENSORFLOW_FOUND)
 endif()
 
 # Perslay
-if(TENSORFLOW_FOUND AND SKLEARN_FOUND)
+if(TENSORFLOW_FOUND AND SCIKIT-LEARN_FOUND)
   add_gudhi_py_test(test_perslay)
 endif()
 
 # Betti curves
-if(SKLEARN_FOUND AND SCIPY_FOUND)
+if(SCIKIT-LEARN_FOUND AND SCIPY_FOUND)
   add_gudhi_py_test(test_betti_curve_representations)
 endif()
 
 # Representations preprocessing
-if(SKLEARN_FOUND)
+if(SCIKIT-LEARN_FOUND)
   add_gudhi_py_test(test_representations_preprocessing)
 endif()
 
@@ -676,13 +676,13 @@ endif()
 add_gudhi_py_test(test_time_delay)
 
 # DTM
-if(SCIPY_FOUND AND SKLEARN_FOUND AND TORCH_FOUND AND HNSWLIB_FOUND AND PYKEOPS_FOUND AND EAGERPY_FOUND)
+if(SCIPY_FOUND AND SCIKIT-LEARN_FOUND AND TORCH_FOUND AND HNSWLIB_FOUND AND PYKEOPS_FOUND AND EAGERPY_FOUND)
   add_gudhi_py_test(test_knn)
   add_gudhi_py_test(test_dtm)
 endif()
 
 # Tomato
-if(SCIPY_FOUND AND SKLEARN_FOUND)
+if(SCIPY_FOUND AND SCIKIT-LEARN_FOUND)
   add_gudhi_py_test(test_tomato)
 endif()
 
@@ -697,7 +697,7 @@ if(SCIPY_FOUND)
 endif()
 
 # Cover complex
-if(SKLEARN_FOUND)
+if(SCIKIT-LEARN_FOUND)
   add_gudhi_py_test(test_cover_complex)
 endif()
 
diff --git a/src/python/doc/installation.rst b/src/python/doc/installation.rst
index c1f7bd2c44..067b733efb 100644
--- a/src/python/doc/installation.rst
+++ b/src/python/doc/installation.rst
@@ -165,26 +165,28 @@ A complete configuration would be :
 
 .. code-block:: none
 
-    Pybind11 version 2.8.1
-    Python version 3.9.7
-    Cython version 0.29.24
-    Pytest version 6.2.5
-    Matplotlib version 3.5.0
-    Numpy version 1.21.4
-    Scipy version 1.7.3
-    Scikit-learn version 1.0.1
-    POT version 0.8.0
-    HNSWlib found
-    PyKeOps version [pyKeOps]: 2.1
+    Python version 3.11.9
+    Pybind11 version 2.12.0
+    Cython version 3.0.10
+    Numpy version 1.24.3
+    Pytest version 8.2.1
+    Matplotlib version 3.9.0
+    Scipy version 1.13.1
+    Scikit-learn version 1.3.2
+    POT version 0.9.4
+    HNSWlib version 0.8.0
+    PyTorch version 2.3.0
+    PyKeOps version 2.2.3
     EagerPy version 0.30.0
-    TensorFlow version 2.7.0
-    Sphinx version 4.3.0
-    Sphinx-paramlinks version 0.5.2
-    pydata_sphinx_theme version 0.13.1
-    NetworkX version 3.0
+    TensorFlow version 2.13.1
+    Sphinx version 7.3.7
+    Sphinx-paramlinks version 0.6.0
+    pydata_sphinx_theme version 0.15.2
+    sphinxcontrib-bibtex version 2.6.2
+    NetworkX version 3.3
     Eigen3 version 3.4.0
-    Boost version 1.74.0
-    CGAL version 5.3
+    Boost version 1.84.0
+    CGAL header only version 5.6.1
     GMP_LIBRARIES = /usr/lib/x86_64-linux-gnu/libgmp.so
     GMPXX_LIBRARIES = /usr/lib/x86_64-linux-gnu/libgmpxx.so
     MPFR_LIBRARIES = /usr/lib/x86_64-linux-gnu/libmpfr.so

From 848d853159083041511b070b85389ebb8cb217a6 Mon Sep 17 00:00:00 2001
From: Vincent Rouvreau <vincent.rouvreau@inria.fr>
Date: Wed, 25 Sep 2024 10:03:02 +0200
Subject: [PATCH 3/6] Specific case for PyKeops on Windows

---
 src/cmake/modules/GUDHI_third_party_libraries.cmake | 12 +++++++++++-
 1 file changed, 11 insertions(+), 1 deletion(-)

diff --git a/src/cmake/modules/GUDHI_third_party_libraries.cmake b/src/cmake/modules/GUDHI_third_party_libraries.cmake
index 18149ebcc2..ffed360461 100644
--- a/src/cmake/modules/GUDHI_third_party_libraries.cmake
+++ b/src/cmake/modules/GUDHI_third_party_libraries.cmake
@@ -179,7 +179,6 @@ if (WITH_GUDHI_PYTHON)
     find_python_module("POT")
     find_python_module("pybind11")
     find_python_module("torch")
-    find_python_module("pykeops")
     find_python_module("eagerpy")
     find_python_module("hnswlib")
     find_python_module("tensorflow")
@@ -189,6 +188,17 @@ if (WITH_GUDHI_PYTHON)
     find_python_module("networkx")
   endif()
 
+  # Specific case for PyKeops that can be imported on Windows, but fails because it uses fcntl (not available on Windows)
+  # Also fcntl has no metadata, so find_python_module does not work
+  # "import fcntl" is about 1 sec. faster than "import pykeops"
+  execute_process(
+          COMMAND ${Python_EXECUTABLE}  -c "import fcntl"
+          RESULT_VARIABLE FCNTL_IMPORT_MODULE_RESULT
+          OUTPUT_VARIABLE FCNTL_IMPORT_MODULE_OUPUT)
+  if(FCNTL_IMPORT_MODULE_RESULT EQUAL 0)
+    find_python_module("pykeops")
+  endif()
+
   if(NOT GUDHI_PYTHON_PATH)
     message(FATAL_ERROR "ERROR: GUDHI_PYTHON_PATH is not valid.")
   endif(NOT GUDHI_PYTHON_PATH)

From dee13f902aacb50fcf024653600af1cf143a38f5 Mon Sep 17 00:00:00 2001
From: hschreiber <hannah.schreiber.k@googlemail.com>
Date: Wed, 25 Sep 2024 11:49:41 +0200
Subject: [PATCH 4/6] rename face as cell in persistence matrix

---
 .../concept/PersistenceMatrixColumn.h         |   4 +-
 .../concept/PersistenceMatrixOptions.h        |   8 +-
 .../doc/Intro_persistence_matrix.h            |   4 +-
 src/Persistence_matrix/include/gudhi/Matrix.h | 148 +++++------
 .../Persistence_matrix/Boundary_matrix.h      |  46 ++--
 .../gudhi/Persistence_matrix/Chain_matrix.h   | 140 +++++-----
 .../Persistence_matrix/Id_to_index_overlay.h  | 250 +++++++++---------
 .../Position_to_index_overlay.h               |  76 +++---
 .../gudhi/Persistence_matrix/RU_matrix.h      |  70 ++---
 .../gudhi/Persistence_matrix/base_pairing.h   |  12 +-
 ... => boundary_cell_position_to_id_mapper.h} |  14 +-
 .../Persistence_matrix/chain_vine_swap.h      |  14 +-
 .../columns/chain_column_extra_properties.h   |   4 +-
 .../matrix_dimension_holders.h                |  16 +-
 .../gudhi/Persistence_matrix/ru_pairing.h     |  12 +-
 .../gudhi/Persistence_matrix/ru_vine_swap.h   |  18 +-
 src/Persistence_matrix/test/pm_matrix_tests.h |   6 +-
 17 files changed, 421 insertions(+), 421 deletions(-)
 rename src/Persistence_matrix/include/gudhi/Persistence_matrix/{boundary_face_position_to_id_mapper.h => boundary_cell_position_to_id_mapper.h} (74%)

diff --git a/src/Persistence_matrix/concept/PersistenceMatrixColumn.h b/src/Persistence_matrix/concept/PersistenceMatrixColumn.h
index 6eaf75f7c9..8ae47e2924 100644
--- a/src/Persistence_matrix/concept/PersistenceMatrixColumn.h
+++ b/src/Persistence_matrix/concept/PersistenceMatrixColumn.h
@@ -86,7 +86,7 @@ class PersistenceMatrixColumn :
   PersistenceMatrixColumn(Column_settings* colSettings = nullptr);
   /**
    * @brief Constructs a column from the given range of @ref Matrix::Entry_representative. If the dimension is stored,
-   * the face is assumed to be simplicial and its dimension to be `nonZeroRowIndices length - 1` or `0`.
+   * the cell is assumed to be simplicial and its dimension to be `nonZeroRowIndices length - 1` or `0`.
    * Otherwise, the dimension should be specified with another constructor.
    * 
    * @tparam Container Range of @ref Matrix::Entry_representative. Assumed to have a %begin(), %end() and %size()
@@ -102,7 +102,7 @@ class PersistenceMatrixColumn :
   /**
    * @brief Constructs a column from the given range of @ref Matrix::Entry_representative such that the rows can be
    * accessed. Each new entry in the column is also inserted in a row using @ref Row_access::insert_entry.
-   * If the dimension is stored, the face is assumed to be simplicial and its dimension to be
+   * If the dimension is stored, the cell is assumed to be simplicial and its dimension to be
    * `nonZeroRowIndices length - 1` or `0`. Otherwise, the dimension should be specified with another constructor.
    * 
    * @tparam Container Range of @ref Matrix::Entry_representative. Assumed to have a %begin(), %end() and %size()
diff --git a/src/Persistence_matrix/concept/PersistenceMatrixOptions.h b/src/Persistence_matrix/concept/PersistenceMatrixOptions.h
index b4d72dc82b..9d5cc83540 100644
--- a/src/Persistence_matrix/concept/PersistenceMatrixOptions.h
+++ b/src/Persistence_matrix/concept/PersistenceMatrixOptions.h
@@ -93,14 +93,14 @@ struct PersistenceMatrixOptions
    * If set to false, the container is a std::vector. By default, it is recommended to set it to false, but some 
    * methods require it to be true to be enabled: 
    * - @ref Matrix::remove_column(Index) "remove_column(Index)" for @ref basematrix "base matrices",
-   * - @ref Matrix::remove_maximal_face(Index) "remove_maximal_face(Index)" for @ref chainmatrix "chain matrices",
-   * - @ref Matrix::remove_maximal_face(Index, const std::vector<Index>&)
-   *  "remove_maximal_face(ID_index, const std::vector<ID_index>&)" for @ref chainmatrix "chain matrices",
+   * - @ref Matrix::remove_maximal_cell(Index) "remove_maximal_cell(Index)" for @ref chainmatrix "chain matrices",
+   * - @ref Matrix::remove_maximal_cell(Index, const std::vector<Index>&)
+   *  "remove_maximal_cell(ID_index, const std::vector<ID_index>&)" for @ref chainmatrix "chain matrices",
    * - @ref Matrix::remove_last "remove_last()" for @ref chainmatrix "chain matrices" if @ref has_vine_update is true.
    */
   static const bool has_map_column_container;
   /**
-   * @brief If set to true, enables the methods @ref Matrix::remove_maximal_face and @ref Matrix::remove_last,
+   * @brief If set to true, enables the methods @ref Matrix::remove_maximal_cell and @ref Matrix::remove_last,
    * except for @ref basematrix "base matrices" when @ref has_column_compression is true.
    */
   static const bool has_removable_columns;
diff --git a/src/Persistence_matrix/doc/Intro_persistence_matrix.h b/src/Persistence_matrix/doc/Intro_persistence_matrix.h
index 77b2657414..1185655b26 100644
--- a/src/Persistence_matrix/doc/Intro_persistence_matrix.h
+++ b/src/Persistence_matrix/doc/Intro_persistence_matrix.h
@@ -30,12 +30,12 @@ namespace persistence_matrix {
  * The main functionalities are:
  * @li column and row access,
  * @li column addition and scalar multiplication,
- * @li removal of maximal faces while maintaining a valid reduced boundary matrix or compatible chain complex base
+ * @li removal of maximal cells while maintaining a valid reduced boundary matrix or compatible chain complex base
  * and a valid barcode with respect to the new filtration,
  * @li computation of persistent homology (but note that if the barcode is your only necessity, using the
  * @ref persistent_cohomology module is often more performant),
  * @li computation of representative cycles for the cycle classes,
- * @li swapping of two consecutive faces in a filtration (cf. vineyards @cite vineyards) while maintaining a valid
+ * @li swapping of two consecutive cells in a filtration (cf. vineyards @cite vineyards) while maintaining a valid
  * reduced boundary matrix or compatible chain complex base and a valid barcode with respect to the new filtration,
  * 
  * \note Matrix API is in a beta version and may change in incompatible ways in the near future.
diff --git a/src/Persistence_matrix/include/gudhi/Matrix.h b/src/Persistence_matrix/include/gudhi/Matrix.h
index a403ccaa81..d917bddc80 100644
--- a/src/Persistence_matrix/include/gudhi/Matrix.h
+++ b/src/Persistence_matrix/include/gudhi/Matrix.h
@@ -109,7 +109,7 @@ namespace persistence_matrix {
  * always take the same type of index as input (for optimization purposes). So, to avoid confusion, we will name and
  * define here the different possibilities, such that we can directly refer to it in the descriptions of the methods.
  * Note that every column and row in a @ref boundarymatrix "boundary" or @ref chainmatrix "chain matrix" is always
- * associated to a single simplex/face, so in order to avoid repeating formulations like "of the simplex associated to
+ * associated to a single simplex/cell, so in order to avoid repeating formulations like "of the simplex associated to
  * the column" all the time, we will amalgamate both notions together.
  *
  * Let @f$ c @f$ be a column.
@@ -134,7 +134,7 @@ namespace persistence_matrix {
  * indexed by its ID, so it correspond to the @ref IDIdx indexing scheme. If the matrix is not a
  * @ref chainmatrix "chain matrix", @f$ r @f$ will originally also be indexed by the ID, but when a swap occurs,
  * the rows also swap IDs and the new ID has to be used to access @f$ r @f$. This means that when the default
- * @ref IDIdx scheme is used (the faces are numerated in order of appearance in the filtration starting at 0),
+ * @ref IDIdx scheme is used (the cells are numerated in order of appearance in the filtration starting at 0),
  * the indexation of the rows correspond to @ref PosIdx.
  *
  * @tparam PersistenceMatrixOptions Structure encoding all the options of the matrix.
@@ -609,9 +609,9 @@ class Matrix {
    * (as in the implementation of the Zigzag algorithm @cite zigzag for example.)
    *
    * @param birthComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller birth than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller birth than the bar associated to the second one.
    * @param deathComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller death than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller death than the bar associated to the second one.
    */
   Matrix(const std::function<bool(Pos_index,Pos_index)>& birthComparator, 
          const std::function<bool(Pos_index,Pos_index)>& deathComparator);
@@ -633,9 +633,9 @@ class Matrix {
    * method.
    * @param orderedBoundaries Vector of ordered boundaries in filtration order. Indexed continuously starting at 0.
    * @param birthComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller birth than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller birth than the bar associated to the second one.
    * @param deathComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller death than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller death than the bar associated to the second one.
    * @param characteristic Characteristic of the coefficient field. Has to be specified if
    * @ref PersistenceMatrixOptions::is_z2 is false. Default value is 11.
    * Ignored if @ref PersistenceMatrixOptions::is_z2 is true.
@@ -660,9 +660,9 @@ class Matrix {
    *
    * @param numberOfColumns Number of columns to reserve space for.
    * @param birthComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller birth than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller birth than the bar associated to the second one.
    * @param deathComparator Method taking two @ref PosIdx indices as parameter and returns true if and only if the first
-   * face is associated to a bar with strictly smaller death than the bar associated to the second one.
+   * cell is associated to a bar with strictly smaller death than the bar associated to the second one.
    * @param characteristic Characteristic of the coefficient field. If not specified and
    * @ref PersistenceMatrixOptions::is_z2 is false, the characteristic has to be set later with the use of
    * @ref set_characteristic before calling for the first time a method needing it.
@@ -736,10 +736,10 @@ class Matrix {
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary.
    * This means that it is assumed that this method is called on boundaries in the order of the filtration.
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration,
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration,
    * starting at 0. If it is not the case, use the other
-   * @ref insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
-   * instead by indicating the face ID used in the boundaries when the face is inserted.
+   * @ref insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
+   * instead by indicating the cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -759,7 +759,7 @@ class Matrix {
    *
    * @tparam Boundary_range Range of @ref Entry_representative. Assumed to have a begin(), end() and size() method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial,
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial,
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return If it is a @ref chainmatrix "chain matrix", the method returns the @ref MatIdx indices of the unpaired
    * chains used to reduce the boundary. Otherwise, nothing.
@@ -768,26 +768,26 @@ class Matrix {
   Insertion_return insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
    * @brief Only available for @ref mp_matrices "non-basic matrices".
-   * It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * for @ref mp_matrices "non-basic matrices", the faces are inserted by order of filtration), it is sufficient to
-   * indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * for @ref mp_matrices "non-basic matrices", the cells are inserted by order of filtration), it is sufficient to
+   * indicate the ID of the cell being inserted.
    *
    * @tparam Boundary_range Range of @ref Entry_representative. Assumed to have a begin(), end() and size() method.
-   * @param faceIndex @ref IDIdx index to use to identify the new face.
+   * @param cellIndex @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the
-   * @p faceIndex values of precedent calls of the method for the corresponding faces and should be ordered in
+   * @p cellIndex values of precedent calls of the method for the corresponding cells and should be ordered in
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial,
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial,
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return If it is a @ref chainmatrix "chain matrix", the method returns the @ref MatIdx indices of the unpaired
    * chains used to reduce the boundary. Otherwise, nothing.
    */
   template <class Boundary_range = Boundary>
-  Insertion_return insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim = -1);
+  Insertion_return insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim = -1);
 
   /**
    * @brief Returns the column at the given @ref MatIdx index.
@@ -895,10 +895,10 @@ class Matrix {
    * @brief Only available for @ref boundarymatrix "RU" and @ref chainmatrix "chain matrices" and if
    * @ref PersistenceMatrixOptions::has_removable_columns and @ref PersistenceMatrixOptions::has_vine_update are true.
    * For @ref chainmatrix "chain matrices", @ref PersistenceMatrixOptions::has_map_column_container and
-   * @ref PersistenceMatrixOptions::has_column_pairings also need to be true. Assumes that the face is maximal in the
+   * @ref PersistenceMatrixOptions::has_column_pairings also need to be true. Assumes that the cell is maximal in the
    * current complex and removes it such that the matrix remains consistent (i.e., RU is still an upper triangular
    * decomposition of the boundary matrix and chain is still a compatible bases of the chain complex in the sense
-   * of @cite zigzag). The maximality of the face is not verified. Also updates the barcode if it was computed.
+   * of @cite zigzag). The maximality of the cell is not verified. Also updates the barcode if it was computed.
    *
    * For @ref chainmatrix "chain matrices", using the other version of the method could perform better depending on how
    * the data is maintained on the side of the user. Then, @ref PersistenceMatrixOptions::has_column_pairings also do
@@ -906,56 +906,56 @@ class Matrix {
    *
    * See also @ref remove_last and @ref remove_column.
    *
-   * @param columnIndex If @ref boundarymatrix "boundary matrix", @ref MatIdx index of the face to remove, otherwise the
+   * @param columnIndex If @ref boundarymatrix "boundary matrix", @ref MatIdx index of the cell to remove, otherwise the
    * @ref IDIdx index.
    */
-  void remove_maximal_face(Index columnIndex);
+  void remove_maximal_cell(Index columnIndex);
   //TODO: See if it would be better to use something more general than a vector for columnsToSwap, such that
   // the user do not have to construct the vector from scratch. Like passing iterators instead. But it would be nice,
-  // to still be able to do (face, {})...
+  // to still be able to do (cell, {})...
   /**
    * @brief Only available for @ref chainmatrix "chain matrices" and if
    * @ref PersistenceMatrixOptions::has_removable_columns, @ref PersistenceMatrixOptions::has_vine_update and
-   * @ref PersistenceMatrixOptions::has_map_column_container are true. Assumes that the face is maximal in the current
+   * @ref PersistenceMatrixOptions::has_map_column_container are true. Assumes that the cell is maximal in the current
    * complex and removes it such that the matrix remains consistent (i.e., it is still a compatible bases of the chain
-   * complex in the sense of @cite zigzag). The maximality of the face is not verified. Also updates the barcode if it
+   * complex in the sense of @cite zigzag). The maximality of the cell is not verified. Also updates the barcode if it
    * was computed.
    *
-   * To maintain the compatibility, vine swaps are done to move the face up to the end of the filtration. Once at the
+   * To maintain the compatibility, vine swaps are done to move the cell up to the end of the filtration. Once at the
    * end, the removal is trivial. But for @ref chainmatrix "chain matrices", swaps do not actually swap the position of
-   * the column every time, so the faces appearing after @p faceIndex in the filtration have to be searched first within
-   * the matrix. If the user has an easy access to the @ref IDIdx of the faces in the order of filtration, passing them
+   * the column every time, so the cells appearing after @p cellIndex in the filtration have to be searched first within
+   * the matrix. If the user has an easy access to the @ref IDIdx of the cells in the order of filtration, passing them
    * by argument with @p columnsToSwap allows to skip a linear search process. Typically, if the user knows that the
-   * face he wants to remove is already the last face of the filtration, calling
-   * @ref remove_maximal_face(ID_index faceIndex, const std::vector<ID_index>& columnsToSwap)
-   * "remove_maximal_face(faceID, {})" will be faster than @ref remove_last().
+   * cell he wants to remove is already the last cell of the filtration, calling
+   * @ref remove_maximal_cell(ID_index cellIndex, const std::vector<ID_index>& columnsToSwap)
+   * "remove_maximal_cell(cellID, {})" will be faster than @ref remove_last().
    *
    * See also @ref remove_last.
    *
-   * @param faceIndex @ref IDIdx index of the face to remove
-   * @param columnsToSwap Vector of @ref IDIdx indices of the faces coming after @p faceIndex in the filtration.
+   * @param cellIndex @ref IDIdx index of the cell to remove
+   * @param columnsToSwap Vector of @ref IDIdx indices of the cells coming after @p cellIndex in the filtration.
    */
-  void remove_maximal_face(ID_index faceIndex, const std::vector<ID_index>& columnsToSwap);
+  void remove_maximal_cell(ID_index cellIndex, const std::vector<ID_index>& columnsToSwap);
   /**
-   * @brief Removes the last inserted column/face from the matrix.
+   * @brief Removes the last inserted column/cell from the matrix.
    * If the matrix is @ref mp_matrices "non basic", @ref PersistenceMatrixOptions::has_removable_columns has to be true
    * for the method to be available. Additionally, if the matrix is a @ref chainmatrix "chain matrix", either
    * @ref PersistenceMatrixOptions::has_map_column_container has to be true or
    * @ref PersistenceMatrixOptions::has_vine_update has to be false. And if the matrix is a
    * @ref basematrix "base matrix" it should be without column compression.
    *
-   * See also @ref remove_maximal_face and @ref remove_column.
+   * See also @ref remove_maximal_cell and @ref remove_column.
    *
-   * For @ref chainmatrix "chain matrices", if @ref PersistenceMatrixOptions::has_vine_update is true, the last face
+   * For @ref chainmatrix "chain matrices", if @ref PersistenceMatrixOptions::has_vine_update is true, the last cell
    * does not have to be at the end of the matrix and therefore has to be searched first. In this case, if the user
-   * already knows the @ref IDIdx of the last face, calling
-   * @ref remove_maximal_face(ID_index faceIndex, const std::vector<ID_index>& columnsToSwap)
-   * "remove_maximal_face(faceID, {})" instead allows to skip the search.
+   * already knows the @ref IDIdx of the last cell, calling
+   * @ref remove_maximal_cell(ID_index cellIndex, const std::vector<ID_index>& columnsToSwap)
+   * "remove_maximal_cell(cellID, {})" instead allows to skip the search.
    */
   void remove_last();
 
   /**
-   * @brief Returns the maximal dimension of a face stored in the matrix. Only available for
+   * @brief Returns the maximal dimension of a cell stored in the matrix. Only available for
    * @ref mp_matrices "non-basic matrices" and if @ref PersistenceMatrixOptions::has_matrix_maximal_dimension_access
    * is true.
    *
@@ -969,10 +969,10 @@ class Matrix {
    */
   Index get_number_of_columns() const;
   /**
-   * @brief Returns the dimension of the given face. Only available for @ref mp_matrices "non-basic matrices".
+   * @brief Returns the dimension of the given cell. Only available for @ref mp_matrices "non-basic matrices".
    * 
-   * @param columnIndex @ref MatIdx index of the column representing the face.
-   * @return Dimension of the face.
+   * @param columnIndex @ref MatIdx index of the column representing the cell.
+   * @return Dimension of the cell.
    */
   Dimension get_column_dimension(Index columnIndex) const;
 
@@ -1189,10 +1189,10 @@ class Matrix {
    * indices and that the @ref rowindex "row indices" for a @ref boundarymatrix "RU matrix" correspond to the updated
    * @ref IDIdx indices which got potentially swapped by a vine swap.
    *
-   * @param faceIndex @ref rowindex "Row index" of the pivot.
+   * @param cellIndex @ref rowindex "Row index" of the pivot.
    * @return @ref MatIdx index of the column with the given pivot.
    */
-  Index get_column_with_pivot(ID_index faceIndex) const;
+  Index get_column_with_pivot(ID_index cellIndex) const;
   /**
    * @brief Returns the @ref rowindex "row index" of the pivot of the given column. Only available for
    * @ref mp_matrices "non-basic matrices".
@@ -1280,7 +1280,7 @@ class Matrix {
    * Does the same than @ref vine_swap, but assumes that the swap is non trivial and therefore skips a part of the case
    * study.
    *
-   * @param index @ref PosIdx index of the first face to swap. The second one has to be at `index + 1`. Recall that for
+   * @param index @ref PosIdx index of the first cell to swap. The second one has to be at `index + 1`. Recall that for
    * @ref boundarymatrix "boundary matrices", @ref PosIdx == @ref MatIdx.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
@@ -1292,8 +1292,8 @@ class Matrix {
    * @ref Column_indexation_types::IDENTIFIER. Does the same than @ref vine_swap, but assumes that the swap is
    * non-trivial and therefore skips a part of the case study.
    *
-   * @param columnIndex1 @ref MatIdx index of the first face.
-   * @param columnIndex2 @ref MatIdx index of the second face. It is assumed that the @ref PosIdx of both only differs
+   * @param columnIndex1 @ref MatIdx index of the first cell.
+   * @param columnIndex2 @ref MatIdx index of the second cell. It is assumed that the @ref PosIdx of both only differs
    * by one.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
@@ -1303,14 +1303,14 @@ class Matrix {
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_vine_update is true and if it is either a
    * @ref boundarymatrix "boundary matrix" or @ref PersistenceMatrixOptions::column_indexation_type is set to
-   * @ref Column_indexation_types::POSITION. Does a vine swap between two faces which are consecutive in the
+   * @ref Column_indexation_types::POSITION. Does a vine swap between two cells which are consecutive in the
    * filtration. Roughly, if \f$ F \f$ is the current filtration represented by the matrix, the method modifies the
    * matrix such that the new state corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but
-   * with the two faces at position `index` and `index + 1` swapped. Of course, the two faces should not have a
+   * with the two cells at position `index` and `index + 1` swapped. Of course, the two cells should not have a
    * face/coface relation which each other ; \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    *
-   * @param index @ref PosIdx index of the first face to swap. The second one has to be at `index + 1`. Recall that for
+   * @param index @ref PosIdx index of the first cell to swap. The second one has to be at `index + 1`. Recall that for
    * @ref boundarymatrix "boundary matrices", @ref PosIdx == @ref MatIdx.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
@@ -1319,15 +1319,15 @@ class Matrix {
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_vine_update is true and if it is either a
    * @ref chainmatrix "chain matrix" or @ref PersistenceMatrixOptions::column_indexation_type is set to
-   * @ref Column_indexation_types::IDENTIFIER. Does a vine swap between two faces which are consecutive in the
+   * @ref Column_indexation_types::IDENTIFIER. Does a vine swap between two cells which are consecutive in the
    * filtration. Roughly, if \f$ F \f$ is the current filtration represented by the matrix, the method modifies the
    * matrix such that the new state corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but
-   * with the two given faces at swapped positions. Of course, the two faces should not have a face/coface relation
+   * with the two given cells at swapped positions. Of course, the two cells should not have a face/coface relation
    * which each other ; \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    *
-   * @param columnIndex1 @ref MatIdx index of the first face.
-   * @param columnIndex2 @ref MatIdx index of the second face. It is assumed that the @ref PosIdx of both only differs
+   * @param columnIndex1 @ref MatIdx index of the first cell.
+   * @param columnIndex2 @ref MatIdx index of the second cell. It is assumed that the @ref PosIdx of both only differs
    * by one.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
@@ -1531,7 +1531,7 @@ inline void Matrix<PersistenceMatrixOptions>::insert_column(const Container& col
 
   static_assert(
       !isNonBasic,
-      "'insert_column' not available for the chosen options. The input has to be in the form of a face boundary.");
+      "'insert_column' not available for the chosen options. The input has to be in the form of a cell boundary.");
   matrix_.insert_column(column);
 }
 
@@ -1573,7 +1573,7 @@ Matrix<PersistenceMatrixOptions>::insert_boundary(const Boundary_range& boundary
 template <class PersistenceMatrixOptions>
 template <class Boundary_range>
 inline typename Matrix<PersistenceMatrixOptions>::Insertion_return
-Matrix<PersistenceMatrixOptions>::insert_boundary(ID_index faceIndex,
+Matrix<PersistenceMatrixOptions>::insert_boundary(ID_index cellIndex,
                                                   const Boundary_range& boundary,
                                                   Dimension dim)
 {
@@ -1586,9 +1586,9 @@ Matrix<PersistenceMatrixOptions>::insert_boundary(ID_index faceIndex,
   static_assert(isNonBasic, "Only enabled for non-basic matrices.");
   if constexpr (!PersistenceMatrixOptions::is_of_boundary_type &&
                 PersistenceMatrixOptions::column_indexation_type == Column_indexation_types::CONTAINER)
-    return matrix_.insert_boundary(faceIndex, boundary, dim);
+    return matrix_.insert_boundary(cellIndex, boundary, dim);
   else
-    matrix_.insert_boundary(faceIndex, boundary, dim);
+    matrix_.insert_boundary(cellIndex, boundary, dim);
 }
 
 template <class PersistenceMatrixOptions>
@@ -1673,31 +1673,31 @@ inline void Matrix<PersistenceMatrixOptions>::erase_empty_row(ID_index rowIndex)
 }
 
 template <class PersistenceMatrixOptions>
-inline void Matrix<PersistenceMatrixOptions>::remove_maximal_face(Index columnIndex) 
+inline void Matrix<PersistenceMatrixOptions>::remove_maximal_cell(Index columnIndex) 
 {
   static_assert(PersistenceMatrixOptions::has_removable_columns,
-                "'remove_maximal_face(ID_index)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index)' is not available for the chosen options.");
   static_assert(isNonBasic && PersistenceMatrixOptions::has_vine_update,
-                "'remove_maximal_face(ID_index)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index)' is not available for the chosen options.");
   static_assert(PersistenceMatrixOptions::is_of_boundary_type || (PersistenceMatrixOptions::has_map_column_container &&
                                                                   PersistenceMatrixOptions::has_column_pairings),
-                "'remove_maximal_face(ID_index)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index)' is not available for the chosen options.");
 
-  matrix_.remove_maximal_face(columnIndex);
+  matrix_.remove_maximal_cell(columnIndex);
 }
 
 template <class PersistenceMatrixOptions>
-inline void Matrix<PersistenceMatrixOptions>::remove_maximal_face(ID_index faceIndex,
+inline void Matrix<PersistenceMatrixOptions>::remove_maximal_cell(ID_index cellIndex,
                                                                   const std::vector<ID_index>& columnsToSwap)
 {
   static_assert(PersistenceMatrixOptions::has_removable_columns,
-                "'remove_maximal_face(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
   static_assert(isNonBasic && !PersistenceMatrixOptions::is_of_boundary_type,
-                "'remove_maximal_face(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
   static_assert(PersistenceMatrixOptions::has_map_column_container && PersistenceMatrixOptions::has_vine_update,
-                "'remove_maximal_face(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
 
-  matrix_.remove_maximal_face(faceIndex, columnsToSwap);
+  matrix_.remove_maximal_cell(cellIndex, columnsToSwap);
 }
 
 template <class PersistenceMatrixOptions>
@@ -1902,14 +1902,14 @@ inline bool Matrix<PersistenceMatrixOptions>::is_zero_column(Index columnIndex,
 
 template <class PersistenceMatrixOptions>
 inline typename Matrix<PersistenceMatrixOptions>::Index Matrix<PersistenceMatrixOptions>::get_column_with_pivot(
-    ID_index faceIndex) const
+    ID_index cellIndex) const
 {
   static_assert(isNonBasic && (!PersistenceMatrixOptions::is_of_boundary_type ||
                                (PersistenceMatrixOptions::has_vine_update ||
                                 PersistenceMatrixOptions::can_retrieve_representative_cycles)),
                 "'get_column_with_pivot' is not available for the chosen options.");
 
-  return matrix_.get_column_with_pivot(faceIndex);
+  return matrix_.get_column_with_pivot(cellIndex);
 }
 
 template <class PersistenceMatrixOptions>
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Boundary_matrix.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Boundary_matrix.h
index 29c6f87d93..93de613c03 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Boundary_matrix.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Boundary_matrix.h
@@ -32,7 +32,7 @@ namespace persistence_matrix {
  *
  * @brief %Matrix structure to store the ordered @ref boundarymatrix "boundary matrix" \f$ R \f$ of a filtered complex
  * in order to compute its persistent homology. Provides an access to its columns and rows as well as the possibility
- * to remove the last faces of the filtration while maintaining a valid barcode.
+ * to remove the last cells of the filtration while maintaining a valid barcode.
  * 
  * @tparam Master_matrix An instantiation of @ref Matrix from which all types and options are deduced.
  */
@@ -120,10 +120,10 @@ class Boundary_matrix : public Master_matrix::Matrix_dimension_option,
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary. 
    * This means that it is assumed that this method is called on boundaries in the order of the filtration. 
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration, 
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration, 
    * starting at 0. If it is not the case, use the other
-   * @ref insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
-   * instead by indicating the face ID used in the boundaries when the face is inserted.
+   * @ref insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
+   * instead by indicating the cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -136,32 +136,32 @@ class Boundary_matrix : public Master_matrix::Matrix_dimension_option,
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return The @ref MatIdx index of the inserted boundary.
    */
   template <class Boundary_range = Boundary>
   Index insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
-   * @brief It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * @brief It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * the faces are inserted by order of filtration), it is sufficient to indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * the cells are inserted by order of filtration), it is sufficient to indicate the ID of the cell being inserted.
    * 
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
-   * @param faceIndex @ref IDIdx index to use to identify the new face.
+   * @param cellIndex @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the 
-   * @p faceIndex values of precedent calls of the method for the corresponding faces and should be ordered in 
+   * @p cellIndex values of precedent calls of the method for the corresponding cells and should be ordered in 
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return The @ref MatIdx index of the inserted boundary.
    */
   template <class Boundary_range = Boundary>
-  Index insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim = -1);
+  Index insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim = -1);
   /**
    * @brief Returns the column at the given @ref MatIdx index.
    * The type of the column depends on the choosen options, see @ref PersistenceMatrixOptions::column_type.
@@ -189,9 +189,9 @@ class Boundary_matrix : public Master_matrix::Matrix_dimension_option,
   Row& get_row(Index rowIndex);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns is true.
-   * Removes the last face in the filtration from the matrix and updates the barcode if this one was already computed.
+   * Removes the last cell in the filtration from the matrix and updates the barcode if this one was already computed.
    * 
-   * @return The pivot of the removed face.
+   * @return The pivot of the removed cell.
    */
   Index remove_last();
   /**
@@ -221,8 +221,8 @@ class Boundary_matrix : public Master_matrix::Matrix_dimension_option,
   /**
    * @brief Returns the dimension of the given column.
    * 
-   * @param columnIndex @ref MatIdx index of the column representing the face.
-   * @return Dimension of the face.
+   * @param columnIndex @ref MatIdx index of the column representing the cell.
+   * @return Dimension of the cell.
    */
   Dimension get_column_dimension(Index columnIndex) const;
 
@@ -469,7 +469,7 @@ inline typename Boundary_matrix<Master_matrix>::Index Boundary_matrix<Master_mat
 template <class Master_matrix>
 template <class Boundary_range>
 inline typename Boundary_matrix<Master_matrix>::Index Boundary_matrix<Master_matrix>::insert_boundary(
-    ID_index faceIndex, const Boundary_range& boundary, Dimension dim) 
+    ID_index cellIndex, const Boundary_range& boundary, Dimension dim) 
 {
   if (dim == -1) dim = boundary.size() == 0 ? 0 : boundary.size() - 1;
 
@@ -492,12 +492,12 @@ inline typename Boundary_matrix<Master_matrix>::Index Boundary_matrix<Master_mat
   //row swap map containers
   if constexpr (Master_matrix::Option_list::has_map_column_container) {
     if constexpr (activeSwapOption) {
-      Swap_opt::indexToRow_.emplace(faceIndex, faceIndex);
-      Swap_opt::rowToIndex_.emplace(faceIndex, faceIndex);
+      Swap_opt::indexToRow_.emplace(cellIndex, cellIndex);
+      Swap_opt::rowToIndex_.emplace(cellIndex, cellIndex);
     }
   } else {
     if constexpr (activeSwapOption) {
-      for (Index i = Swap_opt::indexToRow_.size(); i <= faceIndex; ++i) {
+      for (Index i = Swap_opt::indexToRow_.size(); i <= cellIndex; ++i) {
         Swap_opt::indexToRow_.push_back(i);
         Swap_opt::rowToIndex_.push_back(i);
       }
@@ -506,10 +506,10 @@ inline typename Boundary_matrix<Master_matrix>::Index Boundary_matrix<Master_mat
 
   //maps for possible shifting between column content and position indices used for birth events
   if constexpr (activePairingOption){
-    if (faceIndex != nextInsertIndex_){
-      Pair_opt::idToPosition_.emplace(faceIndex, nextInsertIndex_);
+    if (cellIndex != nextInsertIndex_){
+      Pair_opt::idToPosition_.emplace(cellIndex, nextInsertIndex_);
       if constexpr (Master_matrix::Option_list::has_removable_columns){
-        Pair_opt::PIDM::map_.emplace(nextInsertIndex_, faceIndex);
+        Pair_opt::PIDM::map_.emplace(nextInsertIndex_, cellIndex);
       }
     }
   }
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Chain_matrix.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Chain_matrix.h
index e113fb61b5..7611990b82 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Chain_matrix.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Chain_matrix.h
@@ -35,9 +35,9 @@ namespace persistence_matrix {
  * @ingroup persistence_matrix
  *
  * @brief %Matrix structure storing a compatible base of a filtered chain complex. See @cite zigzag.
- * The base is constructed from the boundaries of the faces in the complex. Allows the persistent homology to be
+ * The base is constructed from the boundaries of the cells in the complex. Allows the persistent homology to be
  * computed, as well as representative cycles. Supports vineyards (see @cite vineyards) and the removal 
- * of maximal faces while maintaining a valid barcode. Provides an access to its columns and rows.
+ * of maximal cells while maintaining a valid barcode. Provides an access to its columns and rows.
  * 
  * @tparam Master_matrix An instantiation of @ref Matrix from which all types and options are deduced.
  */
@@ -228,10 +228,10 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary.
    * This means that it is assumed that this method is called on boundaries in the order of the filtration.
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration,
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration,
    * starting at 0. If it is not the case, use the other
-   * @ref insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
-   * instead by indicating the face ID used in the boundaries when the face is inserted.
+   * @ref insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim) "insert_boundary"
+   * instead by indicating the cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -242,32 +242,32 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return The @ref MatIdx indices of the unpaired chains used to reduce the boundary.
    */
   template <class Boundary_range = Boundary>
   std::vector<Entry_representative> insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
-   * @brief It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * @brief It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * the faces are inserted by order of filtration), it is sufficient to indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * the cells are inserted by order of filtration), it is sufficient to indicate the ID of the cell being inserted.
    * 
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
-   * @param faceID @ref IDIdx index to use to identify the new face.
+   * @param cellID @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the 
-   * @p faceID values of precedent calls of the method for the corresponding faces and should be ordered in 
+   * @p cellID values of precedent calls of the method for the corresponding cells and should be ordered in 
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    * @return The @ref MatIdx index of the inserted boundary.
    */
   template <class Boundary_range = Boundary>
-  std::vector<Entry_representative> insert_boundary(ID_index faceID,
+  std::vector<Entry_representative> insert_boundary(ID_index cellID,
                                                     const Boundary_range& boundary,
                                                     Dimension dim = -1);
   /**
@@ -290,9 +290,9 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns and
    * @ref PersistenceMatrixOptions::has_vine_update are true, as well as,
    * @ref PersistenceMatrixOptions::has_map_column_container and @ref PersistenceMatrixOptions::has_column_pairings.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., the matrix is still a compatible bases of the chain complex in the sense of @cite zigzag).
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it is stored.
    *
    * Note that using the other version of the method could perform better depending on how the data is 
@@ -300,46 +300,46 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
    *
    * See also @ref remove_last.
    * 
-   * @param faceID @ref IDIdx index of the face to remove
+   * @param cellID @ref IDIdx index of the cell to remove
    */
-  void remove_maximal_face(ID_index faceID);
+  void remove_maximal_cell(ID_index cellID);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns,
    * @ref PersistenceMatrixOptions::has_vine_update and @ref PersistenceMatrixOptions::has_map_column_container
    * are true.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., it is still a compatible bases of the chain complex in the sense of @cite zigzag).
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it is stored.
    *
-   * To maintain the compatibility, vine swaps are done to move the face up to the end of the filtration. Once at 
+   * To maintain the compatibility, vine swaps are done to move the cell up to the end of the filtration. Once at 
    * the end, the removal is trivial. But for @ref chainmatrix "chain matrices", swaps do not actually swap the position
-   * of the column every time, so the faces appearing after @p faceID in the filtration have to be searched first within
-   * the matrix. If the user has an easy access to the @ref IDIdx of the faces in the order of filtration, passing them
+   * of the column every time, so the cells appearing after @p cellID in the filtration have to be searched first within
+   * the matrix. If the user has an easy access to the @ref IDIdx of the cells in the order of filtration, passing them
    * by argument with @p columnsToSwap allows to skip a linear search process. Typically, if the user knows that the
-   * face he wants to remove is already the last face of the filtration, calling
-   * @ref remove_maximal_face(ID_index faceIndex, const std::vector<ID_index>& columnsToSwap)
-   * "remove_maximal_face(faceID, {})" will be faster than @ref remove_last().
+   * cell he wants to remove is already the last cell of the filtration, calling
+   * @ref remove_maximal_cell(ID_index cellIndex, const std::vector<ID_index>& columnsToSwap)
+   * "remove_maximal_cell(cellID, {})" will be faster than @ref remove_last().
    *
    * See also @ref remove_last.
    * 
-   * @param faceID @ref IDIdx index of the face to remove
-   * @param columnsToSwap Vector of @ref IDIdx indices of the faces coming after @p faceID in the filtration.
+   * @param cellID @ref IDIdx index of the cell to remove
+   * @param columnsToSwap Vector of @ref IDIdx indices of the cells coming after @p cellID in the filtration.
    */
-  void remove_maximal_face(ID_index faceID, const std::vector<ID_index>& columnsToSwap);
+  void remove_maximal_cell(ID_index cellID, const std::vector<ID_index>& columnsToSwap);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns is true and,
    * if @ref PersistenceMatrixOptions::has_map_column_container is true or
    * @ref PersistenceMatrixOptions::has_vine_update is false.
-   * Removes the last face in the filtration from the matrix and updates the barcode if it is stored.
+   * Removes the last cell in the filtration from the matrix and updates the barcode if it is stored.
    *
-   * See also @ref remove_maximal_face.
+   * See also @ref remove_maximal_cell.
    *
-   * @warning If @ref PersistenceMatrixOptions::has_vine_update is true, the last face does not have to
+   * @warning If @ref PersistenceMatrixOptions::has_vine_update is true, the last cell does not have to
    * be at the end of the matrix container and therefore has to be searched first. In this case, if the user
-   * already knows the @ref IDIdx of the last face, calling
-   * @ref remove_maximal_face(ID_index faceIndex, const std::vector<ID_index>& columnsToSwap)
-   * "remove_maximal_face(faceID, {})" instead allows to skip the search.
+   * already knows the @ref IDIdx of the last cell, calling
+   * @ref remove_maximal_cell(ID_index cellIndex, const std::vector<ID_index>& columnsToSwap)
+   * "remove_maximal_cell(cellID, {})" instead allows to skip the search.
    */
   void remove_last();
 
@@ -353,8 +353,8 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
   /**
    * @brief Returns the dimension of the given column.
    * 
-   * @param columnIndex @ref MatIdx index of the column representing the face.
-   * @return Dimension of the face.
+   * @param columnIndex @ref MatIdx index of the column representing the cell.
+   * @return Dimension of the cell.
    */
   Dimension get_column_dimension(Index columnIndex) const;
 
@@ -422,10 +422,10 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
   /**
    * @brief Returns the column with given @ref rowindex "row index" as pivot. Assumes that the pivot exists.
    * 
-   * @param faceID @ref rowindex "Row index" of the pivot.
+   * @param cellID @ref rowindex "Row index" of the pivot.
    * @return @ref MatIdx index of the column with the given pivot.
    */
-  Index get_column_with_pivot(ID_index faceID) const;
+  Index get_column_with_pivot(ID_index cellID) const;
   /**
    * @brief Returns the @ref rowindex "row index" of the pivot of the given column.
    * 
@@ -499,10 +499,10 @@ class Chain_matrix : public Master_matrix::Matrix_dimension_option,
   Column_settings* colSettings_;  /**< Entry factory. */
 
   template <class Boundary_range>
-  std::vector<Entry_representative> _reduce_boundary(ID_index faceID, const Boundary_range& boundary, Dimension dim);
+  std::vector<Entry_representative> _reduce_boundary(ID_index cellID, const Boundary_range& boundary, Dimension dim);
   void _reduce_by_G(Tmp_column& column, std::vector<Entry_representative>& chainsInH, Index currentPivot);
   void _reduce_by_F(Tmp_column& column, std::vector<Entry_representative>& chainsInF, Index currentPivot);
-  void _build_from_H(ID_index faceID, Tmp_column& column, std::vector<Entry_representative>& chainsInH);
+  void _build_from_H(ID_index cellID, Tmp_column& column, std::vector<Entry_representative>& chainsInH);
   void _update_largest_death_in_F(const std::vector<Entry_representative>& chainsInF);
   void _insert_chain(const Tmp_column& column, Dimension dimension);
   void _insert_chain(const Tmp_column& column, Dimension dimension, Index pair);
@@ -681,21 +681,21 @@ inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Ma
 template <class Master_matrix>
 template <class Boundary_range>
 inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Master_matrix>::insert_boundary(
-    ID_index faceID, const Boundary_range& boundary, Dimension dim) 
+    ID_index cellID, const Boundary_range& boundary, Dimension dim) 
 {
   if constexpr (!Master_matrix::Option_list::has_map_column_container) {
-    if (pivotToColumnIndex_.size() <= faceID) {
-      pivotToColumnIndex_.resize(faceID * 2 + 1, -1);
+    if (pivotToColumnIndex_.size() <= cellID) {
+      pivotToColumnIndex_.resize(cellID * 2 + 1, -1);
     }
   }
 
   if constexpr (Master_matrix::Option_list::has_vine_update && Master_matrix::Option_list::has_column_pairings) {
     if constexpr (Master_matrix::Option_list::has_map_column_container) {
-      Swap_opt::CP::pivotToPosition_.try_emplace(faceID, _nextPosition());
+      Swap_opt::CP::pivotToPosition_.try_emplace(cellID, _nextPosition());
     } else {
-      if (Swap_opt::CP::pivotToPosition_.size() <= faceID)
+      if (Swap_opt::CP::pivotToPosition_.size() <= cellID)
         Swap_opt::CP::pivotToPosition_.resize(pivotToColumnIndex_.size(), -1);
-      Swap_opt::CP::pivotToPosition_[faceID] = _nextPosition();
+      Swap_opt::CP::pivotToPosition_[cellID] = _nextPosition();
     }
   }
 
@@ -703,7 +703,7 @@ inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Ma
     Dim_opt::update_up(dim == static_cast<Dimension>(-1) ? (boundary.size() == 0 ? 0 : boundary.size() - 1) : dim);
   }
 
-  return _reduce_boundary(faceID, boundary, dim);
+  return _reduce_boundary(cellID, boundary, dim);
 }
 
 template <class Master_matrix>
@@ -728,22 +728,22 @@ inline const typename Chain_matrix<Master_matrix>::Column& Chain_matrix<Master_m
 }
 
 template <class Master_matrix>
-inline void Chain_matrix<Master_matrix>::remove_maximal_face(ID_index faceID)
+inline void Chain_matrix<Master_matrix>::remove_maximal_cell(ID_index cellID)
 {
   static_assert(Master_matrix::Option_list::has_removable_columns,
-                "'remove_maximal_face' is not implemented for the chosen options.");
+                "'remove_maximal_cell' is not implemented for the chosen options.");
   static_assert(Master_matrix::Option_list::has_map_column_container && 
                     Master_matrix::Option_list::has_vine_update &&
                     Master_matrix::Option_list::has_column_pairings,
-                "'remove_maximal_face' is not implemented for the chosen options.");
+                "'remove_maximal_cell' is not implemented for the chosen options.");
 
   // TODO: find simple test to verify that col at columnIndex is maximal even without row access.
 
   const auto& pivotToPosition = Swap_opt::CP::pivotToPosition_;
-  auto it = pivotToPosition.find(faceID);
-  if (it == pivotToPosition.end()) return;  // face does not exists. TODO: put an assert instead?
+  auto it = pivotToPosition.find(cellID);
+  if (it == pivotToPosition.end()) return;  // cell does not exists. TODO: put an assert instead?
   Pos_index startPos = it->second;
-  Index startIndex = pivotToColumnIndex_.at(faceID);
+  Index startIndex = pivotToColumnIndex_.at(cellID);
 
   if (startPos != _nextPosition() - 1) {
     std::vector<Index> colToSwap;
@@ -765,17 +765,17 @@ inline void Chain_matrix<Master_matrix>::remove_maximal_face(ID_index faceID)
 }
 
 template <class Master_matrix>
-inline void Chain_matrix<Master_matrix>::remove_maximal_face(ID_index faceID,
+inline void Chain_matrix<Master_matrix>::remove_maximal_cell(ID_index cellID,
                                                              const std::vector<ID_index>& columnsToSwap)
 {
   static_assert(Master_matrix::Option_list::has_removable_columns,
-                "'remove_maximal_face' is not implemented for the chosen options.");
+                "'remove_maximal_cell' is not implemented for the chosen options.");
   static_assert(Master_matrix::Option_list::has_map_column_container && Master_matrix::Option_list::has_vine_update,
-                "'remove_maximal_face' is not implemented for the chosen options.");
+                "'remove_maximal_cell' is not implemented for the chosen options.");
 
   // TODO: find simple test to verify that col at columnIndex is maximal even without row access.
 
-  Index startIndex = pivotToColumnIndex_.at(faceID);
+  Index startIndex = pivotToColumnIndex_.at(cellID);
 
   for (ID_index i : columnsToSwap) {
     startIndex = Swap_opt::vine_swap(startIndex, pivotToColumnIndex_.at(i));
@@ -797,7 +797,7 @@ inline void Chain_matrix<Master_matrix>::remove_last()
   if constexpr (Master_matrix::Option_list::has_vine_update) {
     // careful: linear because of the search of the last index. It is better to keep track of the @ref IDIdx index
     // of the last column while performing swaps (or the @ref MatIdx with the return values of `vine_swap` + get_pivot)
-    // and then call `remove_maximal_face` with it and an empty `columnsToSwap`.
+    // and then call `remove_maximal_cell` with it and an empty `columnsToSwap`.
 
     ID_index pivot = 0;
     Index colIndex = 0;
@@ -869,12 +869,12 @@ inline bool Chain_matrix<Master_matrix>::is_zero_column(Index columnIndex)
 
 template <class Master_matrix>
 inline typename Chain_matrix<Master_matrix>::Index Chain_matrix<Master_matrix>::get_column_with_pivot(
-    ID_index faceID) const
+    ID_index cellID) const
 {
   if constexpr (Master_matrix::Option_list::has_map_column_container) {
-    return pivotToColumnIndex_.at(faceID);
+    return pivotToColumnIndex_.at(cellID);
   } else {
-    return pivotToColumnIndex_[faceID];
+    return pivotToColumnIndex_[cellID];
   }
 }
 
@@ -959,7 +959,7 @@ inline void Chain_matrix<Master_matrix>::print() const
 template <class Master_matrix>
 template <class Boundary_range>
 inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Master_matrix>::_reduce_boundary(
-    ID_index faceID, const Boundary_range& boundary, Dimension dim)
+    ID_index cellID, const Boundary_range& boundary, Dimension dim)
 {
   Tmp_column column(boundary.begin(), boundary.end());
   if (dim == static_cast<Dimension>(-1)) dim = boundary.begin() == boundary.end() ? 0 : boundary.size() - 1;
@@ -975,9 +975,9 @@ inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Ma
 
   if (boundary.begin() == boundary.end()) {
     if constexpr (Master_matrix::Option_list::is_z2)
-      column.insert(faceID);
+      column.insert(cellID);
     else
-      column.emplace(faceID, 1);
+      column.emplace(cellID, 1);
     _insert_chain(column, dim);
     return chainsInF;
   }
@@ -989,7 +989,7 @@ inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Ma
 
     if (column.empty()) {
       // produce the sum of all col_h in chains_in_H
-      _build_from_H(faceID, column, chainsInH);
+      _build_from_H(cellID, column, chainsInH);
       // create a new cycle (in F) sigma - \sum col_h
       _insert_chain(column, dim);
       return chainsInF;
@@ -1011,8 +1011,8 @@ inline std::vector<typename Master_matrix::Entry_representative> Chain_matrix<Ma
 
   _update_largest_death_in_F(chainsInF);
 
-  // Compute the new column faceID + \sum col_h, for col_h in chains_in_H
-  _build_from_H(faceID, column, chainsInH);
+  // Compute the new column cellID + \sum col_h, for col_h in chains_in_H
+  _build_from_H(cellID, column, chainsInH);
 
   // Create and insert (\sum col_h) + sigma (in H, paired with chain_fp) in matrix_
   if constexpr (Master_matrix::Option_list::is_z2)
@@ -1064,17 +1064,17 @@ inline void Chain_matrix<Master_matrix>::_reduce_by_F(Tmp_column& column,
 }
 
 template <class Master_matrix>
-inline void Chain_matrix<Master_matrix>::_build_from_H(ID_index faceID,
+inline void Chain_matrix<Master_matrix>::_build_from_H(ID_index cellID,
                                                        Tmp_column& column,
                                                        std::vector<Entry_representative>& chainsInH)
 {
   if constexpr (Master_matrix::Option_list::is_z2) {
-    column.insert(faceID);
+    column.insert(cellID);
     for (Index idx_h : chainsInH) {
       _add_to(get_column(idx_h), column, 1u);
     }
   } else {
-    column.emplace(faceID, 1);
+    column.emplace(cellID, 1);
     for (std::pair<Index, Field_element>& idx_h : chainsInH) {
       _add_to(get_column(idx_h.first), column, idx_h.second);
     }
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Id_to_index_overlay.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Id_to_index_overlay.h
index 91b09f2659..be885d182a 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Id_to_index_overlay.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Id_to_index_overlay.h
@@ -220,10 +220,10 @@ class Id_to_index_overlay
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary. 
    * This means that it is assumed that this method is called on boundaries in the order of the filtration. 
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration, 
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration, 
    * starting at 0. If it is not the case, use the other
    * @ref insert_boundary(ID_index, const Boundary_range&, Dimension) "insert_boundary" instead by indicating the
-   * face ID used in the boundaries when the face is inserted.
+   * cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -241,39 +241,39 @@ class Id_to_index_overlay
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
   void insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
-   * @brief It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * @brief It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * the faces are inserted by order of filtration), it is sufficient to indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * the cells are inserted by order of filtration), it is sufficient to indicate the ID of the cell being inserted.
    * 
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
-   * @param faceIndex @ref IDIdx index to use to identify the new face.
+   * @param cellIndex @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the 
-   * @p faceIndex values of precedent calls of the method for the corresponding faces and should be ordered in 
+   * @p cellIndex values of precedent calls of the method for the corresponding cells and should be ordered in 
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
-  void insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim = -1);
+  void insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim = -1);
   /**
    * @brief Returns the column at the given @ref IDIdx index. 
    * For @ref boundarymatrix "RU matrices", the returned column is from \f$ R \f$.
    * The type of the column depends on the choosen options, see @ref PersistenceMatrixOptions::column_type.
    * 
-   * @param faceID @ref IDIdx index of the column to return.
+   * @param cellID @ref IDIdx index of the column to return.
    * @return Reference to the column.
    */
-  Column& get_column(ID_index faceID);
+  Column& get_column(ID_index cellID);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_row_access is true.
    * Returns the row at the given @ref rowindex "row index".
@@ -319,10 +319,10 @@ class Id_to_index_overlay
    * @ref PersistenceMatrixOptions::has_removable_columns and @ref PersistenceMatrixOptions::has_vine_update are true.
    * For @ref chainmatrix "chain matrices", @ref PersistenceMatrixOptions::has_map_column_container and
    * @ref PersistenceMatrixOptions::has_column_pairings also need to be true.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., RU is still an upper triangular decomposition of the @ref boundarymatrix "boundary matrix" and chain is
    * still a compatible bases of the chain complex in the sense of @cite zigzag).
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it was computed.
    *
    * For @ref chainmatrix "chain matrices", using the other version of the method could perform better depending on
@@ -331,51 +331,51 @@ class Id_to_index_overlay
    *
    * See also @ref remove_last.
    * 
-   * @param faceID @ref IDIdx index of the face to remove.
+   * @param cellID @ref IDIdx index of the cell to remove.
    */
-  void remove_maximal_face(ID_index faceID);
+  void remove_maximal_cell(ID_index cellID);
   /**
    * @brief Only available for @ref chainmatrix "chain matrices" and if
    * @ref PersistenceMatrixOptions::has_removable_columns, @ref PersistenceMatrixOptions::has_vine_update
    * and @ref PersistenceMatrixOptions::has_map_column_container are true.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., it is still a compatible bases of the chain complex in the sense of @cite zigzag).
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it was computed.
    *
-   * To maintain the compatibility, vine swaps are done to move the face up to the end of the filtration. Once at 
+   * To maintain the compatibility, vine swaps are done to move the cell up to the end of the filtration. Once at 
    * the end, the removal is trivial. But for @ref chainmatrix "chain matrices", swaps do not actually swap the position
-   * of the column every time, so the faces appearing after @p faceIndex in the filtration have to be searched first
-   * within the matrix. If the user has an easy access to the @ref IDIdx of the faces in the order of filtration,
+   * of the column every time, so the cells appearing after @p cellIndex in the filtration have to be searched first
+   * within the matrix. If the user has an easy access to the @ref IDIdx of the cells in the order of filtration,
    * passing them by argument with @p columnsToSwap allows to skip a linear search process. Typically, if the user knows
-   * that the face he wants to remove is already the last face of the filtration, calling
-   * @ref remove_maximal_face(ID_index, const std::vector<ID_index>&) "remove_maximal_face(faceID, {})"
+   * that the cell he wants to remove is already the last cell of the filtration, calling
+   * @ref remove_maximal_cell(ID_index, const std::vector<ID_index>&) "remove_maximal_cell(cellID, {})"
    * will be faster than @ref remove_last().
    *
    * See also @ref remove_last.
    * 
-   * @param faceID @ref IDIdx index of the face to remove.
-   * @param columnsToSwap Vector of @ref IDIdx indices of the faces coming after @p faceID in the filtration.
+   * @param cellID @ref IDIdx index of the cell to remove.
+   * @param columnsToSwap Vector of @ref IDIdx indices of the cells coming after @p cellID in the filtration.
    */
-  void remove_maximal_face(ID_index faceID, const std::vector<ID_index>& columnsToSwap);
+  void remove_maximal_cell(ID_index cellID, const std::vector<ID_index>& columnsToSwap);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns is true. Additionally, if the
    * matrix is a @ref chainmatrix "chain matrix", either @ref PersistenceMatrixOptions::has_map_column_container has to
    * be true or @ref PersistenceMatrixOptions::has_vine_update has to be false.
-   * Removes the last face in the filtration from the matrix and updates the barcode if it is stored.
+   * Removes the last cell in the filtration from the matrix and updates the barcode if it is stored.
    * 
-   * See also @ref remove_maximal_face.
+   * See also @ref remove_maximal_cell.
    *
-   * For @ref chainmatrix "chain matrices", if @ref PersistenceMatrixOptions::has_vine_update is true, the last face
+   * For @ref chainmatrix "chain matrices", if @ref PersistenceMatrixOptions::has_vine_update is true, the last cell
    * does not have to be at the end of the matrix and therefore has to be searched first. In this case, if the user
-   * already knows the @ref IDIdx of the last face, calling
-   * @ref remove_maximal_face(ID_index, const std::vector<ID_index>&) "remove_maximal_face(faceID, {})"
+   * already knows the @ref IDIdx of the last cell, calling
+   * @ref remove_maximal_cell(ID_index, const std::vector<ID_index>&) "remove_maximal_cell(cellID, {})"
    * instead allows to skip the search.
    */
   void remove_last();
 
   /**
-   * @brief Returns the maximal dimension of a face stored in the matrix. Only available 
+   * @brief Returns the maximal dimension of a cell stored in the matrix. Only available 
    * if @ref PersistenceMatrixOptions::has_matrix_maximal_dimension_access is true.
    * 
    * @return The maximal dimension.
@@ -388,24 +388,24 @@ class Id_to_index_overlay
    */
   Index get_number_of_columns() const;
   /**
-   * @brief Returns the dimension of the given face. Only available for @ref mp_matrices "non-basic matrices".
+   * @brief Returns the dimension of the given cell. Only available for @ref mp_matrices "non-basic matrices".
    * 
-   * @param faceID @ref IDIdx index of the face.
-   * @return Dimension of the face.
+   * @param cellID @ref IDIdx index of the cell.
+   * @return Dimension of the cell.
    */
-  Dimension get_column_dimension(ID_index faceID) const;
+  Dimension get_column_dimension(ID_index cellID) const;
 
   /**
-   * @brief Adds column corresponding to @p sourceFaceID onto the column corresponding to @p targetFaceID.
+   * @brief Adds column corresponding to @p sourceCellID onto the column corresponding to @p targetCellID.
    *
    * @warning They will be no verification to ensure that the addition makes sense for the validity of the matrix.
    * For example, a right-to-left addition could corrupt the computation of the barcode if done blindly.
    * So should be used with care.
    * 
-   * @param sourceFaceID @ref IDIdx index of the source column.
-   * @param targetFaceID @ref IDIdx index of the target column.
+   * @param sourceCellID @ref IDIdx index of the source column.
+   * @param targetCellID @ref IDIdx index of the target column.
    */
-  void add_to(ID_index sourceFaceID, ID_index targetFaceID);
+  void add_to(ID_index sourceCellID, ID_index targetCellID);
   /**
    * @brief Multiplies the target column with the coefficient and then adds the source column to it.
    * That is: `targetColumn = (targetColumn * coefficient) + sourceColumn`.
@@ -414,11 +414,11 @@ class Id_to_index_overlay
    * For example, a right-to-left addition could corrupt the computation of the barcode if done blindly.
    * So should be used with care.
    * 
-   * @param sourceFaceID @ref IDIdx index of the source column.
+   * @param sourceCellID @ref IDIdx index of the source column.
    * @param coefficient Value to multiply.
-   * @param targetFaceID @ref IDIdx index of the target column.
+   * @param targetCellID @ref IDIdx index of the target column.
    */
-  void multiply_target_and_add_to(ID_index sourceFaceID, const Field_element& coefficient, ID_index targetFaceID);
+  void multiply_target_and_add_to(ID_index sourceCellID, const Field_element& coefficient, ID_index targetCellID);
   /**
    * @brief Multiplies the source column with the coefficient before adding it to the target column.
    * That is: `targetColumn += (coefficient * sourceColumn)`. The source column will **not** be modified.
@@ -428,10 +428,10 @@ class Id_to_index_overlay
    * So should be used with care.
    * 
    * @param coefficient Value to multiply.
-   * @param sourceFaceID @ref IDIdx index of the source column.
-   * @param targetFaceID @ref IDIdx index of the target column.
+   * @param sourceCellID @ref IDIdx index of the source column.
+   * @param targetCellID @ref IDIdx index of the target column.
    */
-  void multiply_source_and_add_to(const Field_element& coefficient, ID_index sourceFaceID, ID_index targetFaceID);
+  void multiply_source_and_add_to(const Field_element& coefficient, ID_index sourceCellID, ID_index targetCellID);
 
   /**
    * @brief Zeroes the entry at the given coordinates. Not available for @ref chainmatrix "chain matrices".
@@ -440,10 +440,10 @@ class Id_to_index_overlay
    *
    * For @ref boundarymatrix "RU matrices", zeros only the entry in \f$ R \f$.
    * 
-   * @param faceID @ref IDIdx index of the face corresponding to the column of the entry.
+   * @param cellID @ref IDIdx index of the cell corresponding to the column of the entry.
    * @param rowIndex @ref rowindex "Row index" of the row of the entry.
    */
-  void zero_entry(ID_index faceID, ID_index rowIndex);
+  void zero_entry(ID_index cellID, ID_index rowIndex);
   /**
    * @brief Zeroes the column at the given index. Not available for @ref chainmatrix "chain matrices".
    * In general, should be used with care to not destroy the validity 
@@ -451,20 +451,20 @@ class Id_to_index_overlay
    *
    * For @ref boundarymatrix "RU matrices", zeros only the column in \f$ R \f$.
    * 
-   * @param faceID @ref IDIdx index of the face corresponding to the column.
+   * @param cellID @ref IDIdx index of the cell corresponding to the column.
    */
-  void zero_column(ID_index faceID);
+  void zero_column(ID_index cellID);
   /**
    * @brief Indicates if the entry at given coordinates has value zero.
    *
    * For @ref boundarymatrix "RU matrices", looks into \f$ R \f$.
    * 
-   * @param faceID @ref IDIdx index of the face corresponding to the column of the entry.
+   * @param cellID @ref IDIdx index of the cell corresponding to the column of the entry.
    * @param rowIndex @ref rowindex "Row index" of the row of the entry.
    * @return true If the entry has value zero.
    * @return false Otherwise.
    */
-  bool is_zero_entry(ID_index faceID, ID_index rowIndex) const;
+  bool is_zero_entry(ID_index cellID, ID_index rowIndex) const;
   /**
    * @brief Indicates if the column at given index has value zero.
    *
@@ -473,11 +473,11 @@ class Id_to_index_overlay
    * Note that for @ref chainmatrix "chain matrices", this method should always return false, as a valid
    * @ref chainmatrix "chain matrix" never has empty columns.
    * 
-   * @param faceID @ref IDIdx index of the face corresponding to the column.
+   * @param cellID @ref IDIdx index of the cell corresponding to the column.
    * @return true If the column has value zero.
    * @return false Otherwise.
    */
-  bool is_zero_column(ID_index faceID);
+  bool is_zero_column(ID_index cellID);
 
   /**
    * @brief Returns the @ref IDIdx index of the column which has the given @ref rowindex "row index" as pivot.
@@ -487,17 +487,17 @@ class Id_to_index_overlay
    * the row indices for a @ref boundarymatrix "RU matrix" correspond to the updated @ref IDIdx indices which got
    * potentially swapped by a vine swap.
    * 
-   * @param faceIndex @ref rowindex "Row index" of the pivot.
+   * @param cellIndex @ref rowindex "Row index" of the pivot.
    * @return @ref IDIdx index of the column with the given pivot.
    */
-  ID_index get_column_with_pivot(ID_index faceIndex) const;
+  ID_index get_column_with_pivot(ID_index cellIndex) const;
   /**
    * @brief Returns the @ref rowindex "row index" of the pivot of the given column.
    * 
-   * @param faceID @ref IDIdx index of the face corresponding to the column.
+   * @param cellID @ref IDIdx index of the cell corresponding to the column.
    * @return The @ref rowindex "row index" of the pivot.
    */
-  ID_index get_pivot(ID_index faceID);
+  ID_index get_pivot(ID_index cellID);
 
   /**
    * @brief Resets the matrix to an empty matrix.
@@ -549,10 +549,10 @@ class Id_to_index_overlay
    * Swaps the two given columns. Note that it really just swaps two columns and do not updates
    * anything else, nor performs additions to maintain some properties on the matrix.
    * 
-   * @param faceID1 First column @ref IDIdx index to swap.
-   * @param faceID2 Second column @ref IDIdx index to swap.
+   * @param cellID1 First column @ref IDIdx index to swap.
+   * @param cellID2 Second column @ref IDIdx index to swap.
    */
-  void swap_columns(ID_index faceID1, ID_index faceID2);
+  void swap_columns(ID_index cellID1, ID_index cellID2);
   /**
    * @brief Only available for simple @ref boundarymatrix "boundary matrices" (only storing R)
    * and if @ref PersistenceMatrixOptions::has_column_and_row_swaps is true.
@@ -568,29 +568,29 @@ class Id_to_index_overlay
    * Does the same than @ref vine_swap, but assumes that the swap is non trivial and
    * therefore skips a part of the case study.
    * 
-   * @param faceID1 @ref IDIdx index of the first face.
-   * @param faceID2 @ref IDIdx index of the second face. It is assumed that the @ref PosIdx of both only differs by one.
+   * @param cellID1 @ref IDIdx index of the first cell.
+   * @param cellID2 @ref IDIdx index of the second cell. It is assumed that the @ref PosIdx of both only differs by one.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
    * \f$ max(pos1, pos2) \f$.
    */
-  ID_index vine_swap_with_z_eq_1_case(ID_index faceID1, ID_index faceID2);
+  ID_index vine_swap_with_z_eq_1_case(ID_index cellID1, ID_index cellID2);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_vine_update is true.
-   * Does a vine swap between two faces which are consecutive in the filtration. Roughly, if \f$ F \f$ is
+   * Does a vine swap between two cells which are consecutive in the filtration. Roughly, if \f$ F \f$ is
    * the current filtration represented by the matrix, the method modifies the matrix such that the new state
-   * corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two given faces
-   * at swapped positions. Of course, the two faces should not have a face/coface relation which each other ;
+   * corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two given cells
+   * at swapped positions. Of course, the two cells should not have a face/coface relation which each other ;
    * \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    * 
-   * @param faceID1 @ref IDIdx index of the first face.
-   * @param faceID2 @ref IDIdx index of the second face. It is assumed that the @ref PosIdx of both only differs by one.
+   * @param cellID1 @ref IDIdx index of the first cell.
+   * @param cellID2 @ref IDIdx index of the second cell. It is assumed that the @ref PosIdx of both only differs by one.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
    * \f$ max(pos1, pos2) \f$.
    */
-  ID_index vine_swap(ID_index faceID1, ID_index faceID2);
+  ID_index vine_swap(ID_index cellID1, ID_index cellID2);
   
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::can_retrieve_representative_cycles is true. Pre-computes
@@ -752,26 +752,26 @@ inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::insert_bounda
 
 template <class Underlying_matrix, class Master_matrix>
 template <class Boundary_range>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::insert_boundary(ID_index faceIndex,
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::insert_boundary(ID_index cellIndex,
                                                                                   const Boundary_range& boundary,
                                                                                   Dimension dim) 
 {
   if constexpr (Master_matrix::Option_list::has_map_column_container) {
-    GUDHI_CHECK(idToIndex_->find(faceIndex) == idToIndex_->end(),
+    GUDHI_CHECK(idToIndex_->find(cellIndex) == idToIndex_->end(),
                 std::invalid_argument("Id_to_index_overlay::insert_boundary - Index for simplex already chosen!"));
   } else {
-    GUDHI_CHECK((idToIndex_->size() <= faceIndex || _id_to_index(faceIndex) == static_cast<Index>(-1)),
+    GUDHI_CHECK((idToIndex_->size() <= cellIndex || _id_to_index(cellIndex) == static_cast<Index>(-1)),
                 std::invalid_argument("Id_to_index_overlay::insert_boundary - Index for simplex already chosen!"));
   }
-  matrix_.insert_boundary(faceIndex, boundary, dim);
+  matrix_.insert_boundary(cellIndex, boundary, dim);
   if constexpr (Master_matrix::Option_list::is_of_boundary_type) {
     if constexpr (Master_matrix::Option_list::has_map_column_container) {
-      idToIndex_->emplace(faceIndex, nextIndex_);
+      idToIndex_->emplace(cellIndex, nextIndex_);
     } else {
-      if (idToIndex_->size() <= faceIndex) {
-        idToIndex_->resize(faceIndex + 1, -1);
+      if (idToIndex_->size() <= cellIndex) {
+        idToIndex_->resize(cellIndex + 1, -1);
       }
-      _id_to_index(faceIndex) = nextIndex_;
+      _id_to_index(cellIndex) = nextIndex_;
     }
     ++nextIndex_;
   }
@@ -779,9 +779,9 @@ inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::insert_bounda
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Id_to_index_overlay<Underlying_matrix, Master_matrix>::Column&
-Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_column(ID_index faceID) 
+Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_column(ID_index cellID) 
 {
-  return matrix_.get_column(_id_to_index(faceID));
+  return matrix_.get_column(_id_to_index(cellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -798,7 +798,7 @@ inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::erase_empty_r
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_face(ID_index faceID) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_cell(ID_index cellID) 
 {
   if constexpr (Master_matrix::Option_list::is_of_boundary_type) {
     std::vector<ID_index> indexToID(nextIndex_);
@@ -812,34 +812,34 @@ inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maxima
       }
     }
     --nextIndex_;
-    for (Index curr = _id_to_index(faceID); curr < nextIndex_; ++curr) {
+    for (Index curr = _id_to_index(cellID); curr < nextIndex_; ++curr) {
       matrix_.vine_swap(curr);
       std::swap(idToIndex_->at(indexToID[curr]), idToIndex_->at(indexToID[curr + 1]));
     }
     matrix_.remove_last();
-    GUDHI_CHECK(_id_to_index(faceID) == nextIndex_,
-                std::logic_error("Id_to_index_overlay::remove_maximal_face - Indexation problem."));
+    GUDHI_CHECK(_id_to_index(cellID) == nextIndex_,
+                std::logic_error("Id_to_index_overlay::remove_maximal_cell - Indexation problem."));
 
     if constexpr (Master_matrix::Option_list::has_map_column_container) {
-      idToIndex_->erase(faceID);
+      idToIndex_->erase(cellID);
     } else {
-      _id_to_index(faceID) = -1;
+      _id_to_index(cellID) = -1;
     }
   } else {
-    matrix_.remove_maximal_face(faceID);
+    matrix_.remove_maximal_cell(cellID);
   }
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_face(
-    ID_index faceID, const std::vector<ID_index>& columnsToSwap) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_cell(
+    ID_index cellID, const std::vector<ID_index>& columnsToSwap) 
 {
   static_assert(!Master_matrix::Option_list::is_of_boundary_type,
-                "'remove_maximal_face(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
+                "'remove_maximal_cell(ID_index,const std::vector<Index>&)' is not available for the chosen options.");
   std::vector<Index> translatedIndices;
   std::transform(columnsToSwap.cbegin(), columnsToSwap.cend(), std::back_inserter(translatedIndices),
                  [&](ID_index id) { return _id_to_index(id); });
-  matrix_.remove_maximal_face(faceID, translatedIndices);
+  matrix_.remove_maximal_cell(cellID, translatedIndices);
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -881,54 +881,54 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_number_of_columns() c
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Id_to_index_overlay<Underlying_matrix, Master_matrix>::Dimension
-Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_column_dimension(ID_index faceID) const 
+Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_column_dimension(ID_index cellID) const 
 {
-  return matrix_.get_column_dimension(_id_to_index(faceID));
+  return matrix_.get_column_dimension(_id_to_index(cellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::add_to(ID_index sourceFaceID, ID_index targetFaceID) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::add_to(ID_index sourceCellID, ID_index targetCellID) 
 {
-  return matrix_.add_to(_id_to_index(sourceFaceID), _id_to_index(targetFaceID));
+  return matrix_.add_to(_id_to_index(sourceCellID), _id_to_index(targetCellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
 inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::multiply_target_and_add_to(
-    ID_index sourceFaceID, const Field_element& coefficient, ID_index targetFaceID) 
+    ID_index sourceCellID, const Field_element& coefficient, ID_index targetCellID) 
 {
-  return matrix_.multiply_target_and_add_to(_id_to_index(sourceFaceID), coefficient, _id_to_index(targetFaceID));
+  return matrix_.multiply_target_and_add_to(_id_to_index(sourceCellID), coefficient, _id_to_index(targetCellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
 inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::multiply_source_and_add_to(
-    const Field_element& coefficient, ID_index sourceFaceID, ID_index targetFaceID) 
+    const Field_element& coefficient, ID_index sourceCellID, ID_index targetCellID) 
 {
-  return matrix_.multiply_source_and_add_to(coefficient, _id_to_index(sourceFaceID), _id_to_index(targetFaceID));
+  return matrix_.multiply_source_and_add_to(coefficient, _id_to_index(sourceCellID), _id_to_index(targetCellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::zero_entry(ID_index faceID, ID_index rowIndex) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::zero_entry(ID_index cellID, ID_index rowIndex) 
 {
-  return matrix_.zero_entry(_id_to_index(faceID), rowIndex);
+  return matrix_.zero_entry(_id_to_index(cellID), rowIndex);
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::zero_column(ID_index faceID) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::zero_column(ID_index cellID) 
 {
-  return matrix_.zero_column(_id_to_index(faceID));
+  return matrix_.zero_column(_id_to_index(cellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline bool Id_to_index_overlay<Underlying_matrix, Master_matrix>::is_zero_entry(ID_index faceID,
+inline bool Id_to_index_overlay<Underlying_matrix, Master_matrix>::is_zero_entry(ID_index cellID,
                                                                                ID_index rowIndex) const 
 {
-  return matrix_.is_zero_entry(_id_to_index(faceID), rowIndex);
+  return matrix_.is_zero_entry(_id_to_index(cellID), rowIndex);
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline bool Id_to_index_overlay<Underlying_matrix, Master_matrix>::is_zero_column(ID_index faceID) 
+inline bool Id_to_index_overlay<Underlying_matrix, Master_matrix>::is_zero_column(ID_index cellID) 
 {
-  return matrix_.is_zero_column(_id_to_index(faceID));
+  return matrix_.is_zero_column(_id_to_index(cellID));
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -947,12 +947,12 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_column_with_pivot(ID_
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Id_to_index_overlay<Underlying_matrix, Master_matrix>::ID_index
-Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_pivot(ID_index faceID) 
+Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_pivot(ID_index cellID) 
 {
   if constexpr (Master_matrix::Option_list::is_of_boundary_type) {
-    return matrix_.get_pivot(_id_to_index(faceID));
+    return matrix_.get_pivot(_id_to_index(cellID));
   } else {
-    return faceID;
+    return cellID;
   }
 }
 
@@ -1004,10 +1004,10 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::get_representative_cycle(
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::swap_columns(ID_index faceID1, ID_index faceID2) 
+inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::swap_columns(ID_index cellID1, ID_index cellID2) 
 {
-  matrix_.swap_columns(_id_to_index(faceID1), _id_to_index(faceID2));
-  std::swap(idToIndex_->at(faceID1), idToIndex_->at(faceID2));
+  matrix_.swap_columns(_id_to_index(cellID1), _id_to_index(cellID2));
+  std::swap(idToIndex_->at(cellID1), idToIndex_->at(cellID2));
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -1018,10 +1018,10 @@ inline void Id_to_index_overlay<Underlying_matrix, Master_matrix>::swap_rows(Ind
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Id_to_index_overlay<Underlying_matrix, Master_matrix>::ID_index
-Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap_with_z_eq_1_case(ID_index faceID1, ID_index faceID2) 
+Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap_with_z_eq_1_case(ID_index cellID1, ID_index cellID2) 
 {
-  Index first = _id_to_index(faceID1);
-  Index second = _id_to_index(faceID2);
+  Index first = _id_to_index(cellID1);
+  Index second = _id_to_index(cellID2);
   if (first > second) std::swap(first, second);
 
   if constexpr (Master_matrix::Option_list::is_of_boundary_type) {
@@ -1031,12 +1031,12 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap_with_z_eq_1_cas
 
     bool change = matrix_.vine_swap_with_z_eq_1_case(first);
 
-    std::swap(idToIndex_->at(faceID1), idToIndex_->at(faceID2));
+    std::swap(idToIndex_->at(cellID1), idToIndex_->at(cellID2));
 
     if (change) {
-      return faceID1;
+      return cellID1;
     }
-    return faceID2;
+    return cellID2;
   } else {
     return matrix_.vine_swap_with_z_eq_1_case(first, second);
   }
@@ -1044,10 +1044,10 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap_with_z_eq_1_cas
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Id_to_index_overlay<Underlying_matrix, Master_matrix>::ID_index
-Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap(ID_index faceID1, ID_index faceID2) 
+Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap(ID_index cellID1, ID_index cellID2) 
 {
-  Index first = _id_to_index(faceID1);
-  Index second = _id_to_index(faceID2);
+  Index first = _id_to_index(cellID1);
+  Index second = _id_to_index(cellID2);
   if (first > second) std::swap(first, second);
 
   if constexpr (Master_matrix::Option_list::is_of_boundary_type) {
@@ -1056,12 +1056,12 @@ Id_to_index_overlay<Underlying_matrix, Master_matrix>::vine_swap(ID_index faceID
 
     bool change = matrix_.vine_swap(first);
 
-    std::swap(idToIndex_->at(faceID1), idToIndex_->at(faceID2));
+    std::swap(idToIndex_->at(cellID1), idToIndex_->at(cellID2));
 
     if (change) {
-      return faceID1;
+      return cellID1;
     }
-    return faceID2;
+    return cellID2;
   } else {
     return matrix_.vine_swap(first, second);
   }
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Position_to_index_overlay.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Position_to_index_overlay.h
index 56cd2c2d27..4a19488230 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/Position_to_index_overlay.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/Position_to_index_overlay.h
@@ -216,10 +216,10 @@ class Position_to_index_overlay
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary. 
    * This means that it is assumed that this method is called on boundaries in the order of the filtration. 
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration, 
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration, 
    * starting at 0. If it is not the case, use the other
    * @ref insert_boundary(ID_index, const Boundary_range&, Dimension) "insert_boundary" instead by indicating the
-   * face ID used in the boundaries when the face is inserted.
+   * cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -230,30 +230,30 @@ class Position_to_index_overlay
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
   void insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
-   * @brief It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * @brief It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * the faces are inserted by order of filtration), it is sufficient to indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * the cells are inserted by order of filtration), it is sufficient to indicate the ID of the cell being inserted.
    * 
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
-   * @param faceIndex @ref IDIdx index to use to identify the new face.
+   * @param cellIndex @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the 
-   * @p faceID values of precedent calls of the method for the corresponding faces and should be ordered in 
+   * @p cellID values of precedent calls of the method for the corresponding cells and should be ordered in 
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial, 
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial, 
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
-  void insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim = -1);
+  void insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim = -1);
   /**
    * @brief Returns the column at the given @ref PosIdx index.
    * The type of the column depends on the choosen options, see @ref PersistenceMatrixOptions::column_type.
@@ -307,28 +307,28 @@ class Position_to_index_overlay
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns,
    * @ref PersistenceMatrixOptions::has_vine_update and @ref PersistenceMatrixOptions::has_map_column_container
    * are true.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., the matrix is still a compatible bases of the chain complex in the sense of @cite zigzag).
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it was computed.
    *
    * See also @ref remove_last.
    * 
-   * @param position @ref PosIdx index of the face to remove.
+   * @param position @ref PosIdx index of the cell to remove.
    */
-  void remove_maximal_face(Pos_index position);
+  void remove_maximal_cell(Pos_index position);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns is true and,
    * if @ref PersistenceMatrixOptions::has_map_column_container is true or
    * @ref PersistenceMatrixOptions::has_vine_update is false.
-   * Removes the last face in the filtration from the matrix and updates the barcode if it is stored.
+   * Removes the last cell in the filtration from the matrix and updates the barcode if it is stored.
    *
-   * See also @ref remove_maximal_face.
+   * See also @ref remove_maximal_cell.
    */
   void remove_last();
 
   /**
-   * @brief Returns the maximal dimension of a face stored in the matrix. Only available 
+   * @brief Returns the maximal dimension of a cell stored in the matrix. Only available 
    * if @ref PersistenceMatrixOptions::has_matrix_maximal_dimension_access is true.
    * 
    * @return The maximal dimension.
@@ -341,10 +341,10 @@ class Position_to_index_overlay
    */
   Index get_number_of_columns() const;
   /**
-   * @brief Returns the dimension of the given face.
+   * @brief Returns the dimension of the given cell.
    * 
-   * @param position @ref PosIdx index of the face.
-   * @return Dimension of the face.
+   * @param position @ref PosIdx index of the cell.
+   * @return Dimension of the cell.
    */
   Dimension get_column_dimension(Pos_index position) const;
 
@@ -393,7 +393,7 @@ class Position_to_index_overlay
   /**
    * @brief Indicates if the entry at given coordinates has value zero.
    * 
-   * @param position @ref PosIdx index of the face corresponding to the column of the entry.
+   * @param position @ref PosIdx index of the cell corresponding to the column of the entry.
    * @param rowIndex @ref rowindex "Row index" of the row of the entry.
    * @return true If the entry has value zero.
    * @return false Otherwise.
@@ -405,7 +405,7 @@ class Position_to_index_overlay
    * Note that this method should always return false, as a valid @ref chainmatrix "chain matrix" never has
    * empty columns.
    * 
-   * @param position @ref PosIdx index of the face corresponding to the column.
+   * @param position @ref PosIdx index of the cell corresponding to the column.
    * @return true If the column has value zero.
    * @return false Otherwise.
    */
@@ -415,14 +415,14 @@ class Position_to_index_overlay
    * @brief Returns the @ref PosIdx index of the column which has the given @ref rowindex "row index" as pivot.
    * Assumes that the pivot exists.
    * 
-   * @param faceIndex @ref rowindex "Row index" of the pivot.
+   * @param cellIndex @ref rowindex "Row index" of the pivot.
    * @return @ref PosIdx index of the column with the given pivot.
    */
-  Pos_index get_column_with_pivot(ID_index faceIndex) const;  // assumes that pivot exists
+  Pos_index get_column_with_pivot(ID_index cellIndex) const;  // assumes that pivot exists
   /**
    * @brief Returns the @ref rowindex "row index" of the pivot of the given column.
    * 
-   * @param position @ref PosIdx index of the face corresponding to the column.
+   * @param position @ref PosIdx index of the cell corresponding to the column.
    * @return The @ref rowindex "row index" of the pivot.
    */
   ID_index get_pivot(Pos_index position);
@@ -500,21 +500,21 @@ class Position_to_index_overlay
    * Does the same than @ref vine_swap, but assumes that the swap is non trivial and
    * therefore skips a part of the case study.
    * 
-   * @param position @ref PosIdx index of the first face to swap. The second one has to be at `position + 1`.
+   * @param position @ref PosIdx index of the first cell to swap. The second one has to be at `position + 1`.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
    */
   bool vine_swap_with_z_eq_1_case(Pos_index position);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_vine_update is true.
-   * Does a vine swap between two faces which are consecutive in the filtration. Roughly, if \f$ F \f$ is the current
+   * Does a vine swap between two cells which are consecutive in the filtration. Roughly, if \f$ F \f$ is the current
    * filtration represented by the matrix, the method modifies the matrix such that the new state corresponds to 
-   * a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two faces at position `position`
-   * and `position + 1` swapped. Of course, the two faces should not have a face/coface relation which each other ;
+   * a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two cells at position `position`
+   * and `position + 1` swapped. Of course, the two cells should not have a face/coface relation which each other ;
    * \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    * 
-   * @param position @ref PosIdx index of the first face to swap. The second one has to be at `position + 1`.
+   * @param position @ref PosIdx index of the first cell to swap. The second one has to be at `position + 1`.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
    */
@@ -629,7 +629,7 @@ inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::insert_
 
 template <class Underlying_matrix, class Master_matrix>
 template <class Boundary_range>
-inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::insert_boundary(ID_index faceIndex,
+inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::insert_boundary(ID_index cellIndex,
                                                                                         const Boundary_range& boundary,
                                                                                         Dimension dim) 
 {
@@ -639,7 +639,7 @@ inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::insert_
 
   positionToIndex_[nextPosition_++] = nextIndex_++;
 
-  matrix_.insert_boundary(faceIndex, boundary, dim);
+  matrix_.insert_boundary(cellIndex, boundary, dim);
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -677,7 +677,7 @@ inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::erase_e
 }
 
 template <class Underlying_matrix, class Master_matrix>
-inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_face(Pos_index position) 
+inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::remove_maximal_cell(Pos_index position) 
 {
   --nextPosition_;
 
@@ -693,7 +693,7 @@ inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::remove_
     columnsToSwap.back() = positionToIndex_[nextPosition_];
   }
 
-  matrix_.remove_maximal_face(pivot, columnsToSwap);
+  matrix_.remove_maximal_cell(pivot, columnsToSwap);
 }
 
 template <class Underlying_matrix, class Master_matrix>
@@ -702,9 +702,9 @@ inline void Position_to_index_overlay<Underlying_matrix, Master_matrix>::remove_
   --nextPosition_;
   if constexpr (Master_matrix::Option_list::has_vine_update) {
     std::vector<Index> columnsToSwap;
-    matrix_.remove_maximal_face(matrix_.get_pivot(positionToIndex_[nextPosition_]), columnsToSwap);
+    matrix_.remove_maximal_cell(matrix_.get_pivot(positionToIndex_[nextPosition_]), columnsToSwap);
   } else {
-    matrix_.remove_last();  // linear with vine updates, so it is better to use remove_maximal_face
+    matrix_.remove_last();  // linear with vine updates, so it is better to use remove_maximal_cell
   }
 }
 
@@ -769,9 +769,9 @@ inline bool Position_to_index_overlay<Underlying_matrix, Master_matrix>::is_zero
 
 template <class Underlying_matrix, class Master_matrix>
 inline typename Position_to_index_overlay<Underlying_matrix, Master_matrix>::Pos_index
-Position_to_index_overlay<Underlying_matrix, Master_matrix>::get_column_with_pivot(ID_index faceIndex) const 
+Position_to_index_overlay<Underlying_matrix, Master_matrix>::get_column_with_pivot(ID_index cellIndex) const 
 {
-  Index id = matrix_.get_column_with_pivot(faceIndex);
+  Index id = matrix_.get_column_with_pivot(cellIndex);
   Pos_index i = 0;
   while (positionToIndex_[i] != id) ++i;
   return i;
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/RU_matrix.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/RU_matrix.h
index f738404e42..7c3661a8a9 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/RU_matrix.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/RU_matrix.h
@@ -30,7 +30,7 @@ namespace persistence_matrix {
  *
  * @brief %Matrix structure to store the ordered @ref boundarymatrix "boundary matrix" \f$ R \cdot U \f$ of a filtered
  * complex in order to compute its persistent homology, as well as representative cycles.
- * Supports vineyards (see @cite vineyards) and the removal of maximal faces while maintaining
+ * Supports vineyards (see @cite vineyards) and the removal of maximal cells while maintaining
  * a valid barcode. Provides an access to its columns and rows.
  *
  * @tparam Master_matrix An instantiation of @ref Matrix from which all types and options are deduced.
@@ -115,10 +115,10 @@ class RU_matrix : public Master_matrix::RU_pairing_option,
   /**
    * @brief Inserts at the end of the matrix a new ordered column corresponding to the given boundary.
    * This means that it is assumed that this method is called on boundaries in the order of the filtration.
-   * It also assumes that the faces in the given boundary are identified by their relative position in the filtration,
+   * It also assumes that the cells in the given boundary are identified by their relative position in the filtration,
    * starting at 0. If it is not the case, use the other
    * @ref insert_boundary(ID_index, const Boundary_range&, Dimension) "insert_boundary" instead by indicating the
-   * face ID used in the boundaries when the face is inserted.
+   * cell ID used in the boundaries when the cell is inserted.
    *
    * Different to the constructor, the boundaries do not have to come from a simplicial complex, but also from
    * a more general entry complex. This includes cubical complexes or Morse complexes for example.
@@ -128,30 +128,30 @@ class RU_matrix : public Master_matrix::RU_pairing_option,
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
    * @param boundary Boundary generating the new column. The content should be ordered by ID.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial,
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial,
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
   void insert_boundary(const Boundary_range& boundary, Dimension dim = -1);
   /**
-   * @brief It does the same as the other version, but allows the boundary faces to be identified without restrictions
+   * @brief It does the same as the other version, but allows the boundary cells to be identified without restrictions
    * except that all IDs have to be strictly increasing in the order of filtration. Note that you should avoid then
    * to use the other insertion method to avoid overwriting IDs.
    *
-   * As a face has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
-   * the faces are inserted by order of filtration), it is sufficient to indicate the ID of the face being inserted.
+   * As a cell has to be inserted before one of its cofaces in a valid filtration (recall that it is assumed that
+   * the cells are inserted by order of filtration), it is sufficient to indicate the ID of the cell being inserted.
    *
    * @tparam Boundary_range Range of @ref Matrix::Entry_representative. Assumed to have a begin(), end() and size()
    * method.
-   * @param faceIndex @ref IDIdx index to use to identify the new face.
+   * @param cellIndex @ref IDIdx index to use to identify the new cell.
    * @param boundary Boundary generating the new column. The indices of the boundary have to correspond to the
-   * @p faceIndex values of precedent calls of the method for the corresponding faces and should be ordered in
+   * @p cellIndex values of precedent calls of the method for the corresponding cells and should be ordered in
    * increasing order.
-   * @param dim Dimension of the face whose boundary is given. If the complex is simplicial,
+   * @param dim Dimension of the cell whose boundary is given. If the complex is simplicial,
    * this parameter can be omitted as it can be deduced from the size of the boundary.
    */
   template <class Boundary_range = Boundary>
-  void insert_boundary(ID_index faceIndex, const Boundary_range& boundary, Dimension dim = -1);
+  void insert_boundary(ID_index cellIndex, const Boundary_range& boundary, Dimension dim = -1);
   /**
    * @brief Returns the column at the given @ref MatIdx index in \f$ R \f$ if @p inR is true and
    * in \f$ U \f$ if @p inR is false.
@@ -203,26 +203,26 @@ class RU_matrix : public Master_matrix::RU_pairing_option,
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns and
    * @ref PersistenceMatrixOptions::has_vine_update are true.
-   * Assumes that the face is maximal in the current complex and removes it such that the matrix remains consistent
+   * Assumes that the cell is maximal in the current complex and removes it such that the matrix remains consistent
    * (i.e., RU is still an upper triangular decomposition of the @ref boundarymatrix "boundary matrix").
-   * The maximality of the face is not verified.
+   * The maximality of the cell is not verified.
    * Also updates the barcode if it is stored.
    *
    * See also @ref remove_last.
    *
-   * @param columnIndex @ref MatIdx index of the face to remove.
+   * @param columnIndex @ref MatIdx index of the cell to remove.
    */
-  void remove_maximal_face(Index columnIndex);
+  void remove_maximal_cell(Index columnIndex);
   /**
    * @brief Only available if @ref PersistenceMatrixOptions::has_removable_columns is true.
-   * Removes the last face in the filtration from the matrix and updates the barcode if it is stored.
+   * Removes the last cell in the filtration from the matrix and updates the barcode if it is stored.
    *
-   * See also @ref remove_maximal_face.
+   * See also @ref remove_maximal_cell.
    */
   void remove_last();
 
   /**
-   * @brief Returns the maximal dimension of a face stored in the matrix.
+   * @brief Returns the maximal dimension of a cell stored in the matrix.
    * Only available if @ref PersistenceMatrixOptions::has_matrix_maximal_dimension_access is true.
    *
    * @return The maximal dimension.
@@ -237,8 +237,8 @@ class RU_matrix : public Master_matrix::RU_pairing_option,
   /**
    * @brief Returns the dimension of the given column.
    *
-   * @param columnIndex @ref MatIdx index of the column representing the face.
-   * @return Dimension of the face.
+   * @param columnIndex @ref MatIdx index of the column representing the cell.
+   * @return Dimension of the cell.
    */
   Dimension get_column_dimension(Index columnIndex) const;
 
@@ -331,10 +331,10 @@ class RU_matrix : public Master_matrix::RU_pairing_option,
    * @brief Returns the @ref MatIdx index of the column which has the given @ref rowindex "row index" as pivot in
    * \f$ R \f$. Assumes that the pivot exists.
    *
-   * @param faceIndex @ref rowindex "Row index" of the pivot.
+   * @param cellIndex @ref rowindex "Row index" of the pivot.
    * @return @ref MatIdx index of the column in \f$ R \f$ with the given pivot.
    */
-  Index get_column_with_pivot(Index faceIndex) const;
+  Index get_column_with_pivot(Index cellIndex) const;
   /**
    * @brief Returns the @ref rowindex "row index" of the pivot of the given column in \f$ R \f$.
    *
@@ -522,28 +522,28 @@ inline void RU_matrix<Master_matrix>::insert_boundary(const Boundary_range& boun
 
 template <class Master_matrix>
 template <class Boundary_range>
-inline void RU_matrix<Master_matrix>::insert_boundary(ID_index faceIndex,
+inline void RU_matrix<Master_matrix>::insert_boundary(ID_index cellIndex,
                                                       const Boundary_range& boundary,
                                                       Dimension dim)
 {
   // maps for possible shifting between column content and position indices used for birth events
   if constexpr (Master_matrix::Option_list::has_column_pairings && !Master_matrix::Option_list::has_vine_update) {
-    if (faceIndex != nextEventIndex_) {
-      Pair_opt::idToPosition_.emplace(faceIndex, nextEventIndex_);
+    if (cellIndex != nextEventIndex_) {
+      Pair_opt::idToPosition_.emplace(cellIndex, nextEventIndex_);
       if constexpr (Master_matrix::Option_list::has_removable_columns) {
-        Pair_opt::PIDM::map_.emplace(nextEventIndex_, faceIndex);
+        Pair_opt::PIDM::map_.emplace(nextEventIndex_, cellIndex);
       }
     }
   }
   if constexpr (Master_matrix::Option_list::has_vine_update) {
-    if (faceIndex != nextEventIndex_) {
-      Swap_opt::_positionToRowIdx().emplace(nextEventIndex_, faceIndex);
+    if (cellIndex != nextEventIndex_) {
+      Swap_opt::_positionToRowIdx().emplace(nextEventIndex_, cellIndex);
       if (Master_matrix::Option_list::has_column_pairings) {
-        Swap_opt::template RU_pairing<Master_matrix>::idToPosition_.emplace(faceIndex, nextEventIndex_);
+        Swap_opt::template RU_pairing<Master_matrix>::idToPosition_.emplace(cellIndex, nextEventIndex_);
       }
     }
   }
-  _insert_boundary(reducedMatrixR_.insert_boundary(faceIndex, boundary, dim));
+  _insert_boundary(reducedMatrixR_.insert_boundary(cellIndex, boundary, dim));
 }
 
 template <class Master_matrix>
@@ -573,10 +573,10 @@ inline void RU_matrix<Master_matrix>::erase_empty_row(Index rowIndex)
 }
 
 template <class Master_matrix>
-inline void RU_matrix<Master_matrix>::remove_maximal_face(Index columnIndex)
+inline void RU_matrix<Master_matrix>::remove_maximal_cell(Index columnIndex)
 {
   static_assert(Master_matrix::Option_list::has_removable_columns && Master_matrix::Option_list::has_vine_update,
-                "'remove_maximal_face' is not implemented for the chosen options.");
+                "'remove_maximal_cell' is not implemented for the chosen options.");
 
   // TODO: is there an easy test to verify maximality even without row access?
 
@@ -699,12 +699,12 @@ inline bool RU_matrix<Master_matrix>::is_zero_column(Index columnIndex, bool inR
 }
 
 template <class Master_matrix>
-inline typename RU_matrix<Master_matrix>::Index RU_matrix<Master_matrix>::get_column_with_pivot(Index faceIndex) const
+inline typename RU_matrix<Master_matrix>::Index RU_matrix<Master_matrix>::get_column_with_pivot(Index cellIndex) const
 {
   if constexpr (Master_matrix::Option_list::has_map_column_container) {
-    return pivotToColumnIndex_.at(faceIndex);
+    return pivotToColumnIndex_.at(cellIndex);
   } else {
-    return pivotToColumnIndex_[faceIndex];
+    return pivotToColumnIndex_[cellIndex];
   }
 }
 
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/base_pairing.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/base_pairing.h
index ac883fff38..8ce95ce743 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/base_pairing.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/base_pairing.h
@@ -23,7 +23,7 @@
 #include <algorithm>
 #include <vector>
 
-#include "boundary_face_position_to_id_mapper.h"
+#include "boundary_cell_position_to_id_mapper.h"
 
 namespace Gudhi {
 namespace persistence_matrix {
@@ -50,7 +50,7 @@ struct Dummy_base_pairing {
 template <class Master_matrix>
 class Base_pairing : public std::conditional<
                        Master_matrix::Option_list::has_removable_columns,
-                       Face_position_to_ID_mapper<typename Master_matrix::ID_index, typename Master_matrix::Pos_index>,
+                       Cell_position_to_ID_mapper<typename Master_matrix::ID_index, typename Master_matrix::Pos_index>,
                        Dummy_pos_mapper
                     >::type
 {
@@ -82,8 +82,8 @@ class Base_pairing : public std::conditional<
    */
   friend void swap(Base_pairing& pairing1, Base_pairing& pairing2) {
     if constexpr (Master_matrix::Option_list::has_removable_columns) {
-      swap(static_cast<Face_position_to_ID_mapper<ID_index, Pos_index>&>(pairing1),
-           static_cast<Face_position_to_ID_mapper<ID_index, Pos_index>&>(pairing2));
+      swap(static_cast<Cell_position_to_ID_mapper<ID_index, Pos_index>&>(pairing1),
+           static_cast<Cell_position_to_ID_mapper<ID_index, Pos_index>&>(pairing2));
     }
     pairing1.barcode_.swap(pairing2.barcode_);
     pairing1.deathToBar_.swap(pairing2.deathToBar_);
@@ -98,14 +98,14 @@ class Base_pairing : public std::conditional<
   using Base_matrix = typename Master_matrix::Master_boundary_matrix;
   //PIDM = Position to ID Map
   using PIDM = typename std::conditional<Master_matrix::Option_list::has_removable_columns,
-                                         Face_position_to_ID_mapper<ID_index, Pos_index>,
+                                         Cell_position_to_ID_mapper<ID_index, Pos_index>,
                                          Dummy_pos_mapper
                                         >::type;
 
   Barcode barcode_;       /**< Bar container. */
   Dictionary deathToBar_; /**< Map from death index to bar index. */
   /**
-   * @brief Map from face ID to face position. Only stores a pair if ID != position.
+   * @brief Map from cell ID to cell position. Only stores a pair if ID != position.
    */
   std::unordered_map<ID_index,Pos_index> idToPosition_;  //TODO: test other map types
   bool isReduced_;        /**< True if `_reduce()` was called. */
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_face_position_to_id_mapper.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_cell_position_to_id_mapper.h
similarity index 74%
rename from src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_face_position_to_id_mapper.h
rename to src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_cell_position_to_id_mapper.h
index 5464804196..1b85b07c82 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_face_position_to_id_mapper.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/boundary_cell_position_to_id_mapper.h
@@ -9,10 +9,10 @@
  */
 
 /**
- * @file base_pairing.h
+ * @file boundary_cell_position_to_id_mapper.h
  * @author Hannah Schreiber
- * @brief Contains the @ref Gudhi::persistence_matrix::Base_pairing class and
- * @ref Gudhi::persistence_matrix::Dummy_base_pairing structure.
+ * @brief Contains the @ref Gudhi::persistence_matrix::Cell_position_to_ID_mapper class and
+ * @ref Gudhi::persistence_matrix::Dummy_pos_mapper structure.
  */
 
 #ifndef PM_ID_POS_MAPPER_H
@@ -28,7 +28,7 @@ namespace persistence_matrix {
  * @ingroup persistence_matrix
  *
  * @brief Empty structure.
- * Inherited instead of @ref Face_position_to_ID_mapper.
+ * Inherited instead of @ref Cell_position_to_ID_mapper.
  */
 struct Dummy_pos_mapper {
   friend void swap([[maybe_unused]] Dummy_pos_mapper& d1, [[maybe_unused]] Dummy_pos_mapper& d2) {}
@@ -38,18 +38,18 @@ struct Dummy_pos_mapper {
  * @private
  * @ingroup persistence_matrix
  *
- * @brief Map from face position to face ID. Only stores a pair if ID != position and has_removable_column is true.
+ * @brief Map from cell position to cell ID. Only stores a pair if ID != position and has_removable_column is true.
  * 
  * @tparam ID_index @ref IDIdx index type
  * @tparam Pos_index @ref PosIdx index type
  */
 template<typename ID_index, typename Pos_index>
-struct Face_position_to_ID_mapper {
+struct Cell_position_to_ID_mapper {
   using Index_map = std::unordered_map<Pos_index,ID_index>; //TODO: test other map types
 
   Index_map map_;
 
-  friend void swap(Face_position_to_ID_mapper& mapper1, Face_position_to_ID_mapper& mapper2) {
+  friend void swap(Cell_position_to_ID_mapper& mapper1, Cell_position_to_ID_mapper& mapper2) {
     mapper1.map_.swap(mapper2.map_);
   }
 };
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/chain_vine_swap.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/chain_vine_swap.h
index 887f89cb2c..3168b54031 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/chain_vine_swap.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/chain_vine_swap.h
@@ -281,23 +281,23 @@ class Chain_vine_swap : public std::conditional<Master_matrix::Option_list::has_
    * @brief Does the same than @ref vine_swap, but assumes that the swap is non trivial and
    * therefore skips a part of the case study.
    * 
-   * @param columnIndex1 @ref MatIdx index of the first face.
-   * @param columnIndex2 @ref MatIdx index of the second face.
+   * @param columnIndex1 @ref MatIdx index of the first cell.
+   * @param columnIndex2 @ref MatIdx index of the second cell.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
    * \f$ max(pos1, pos2) \f$.
    */
   Index vine_swap_with_z_eq_1_case(Index columnIndex1, Index columnIndex2);
   /**
-   * @brief Does a vine swap between two faces which are consecutive in the filtration. Roughly, if \f$ F \f$ is
+   * @brief Does a vine swap between two cells which are consecutive in the filtration. Roughly, if \f$ F \f$ is
    * the current filtration represented by the matrix, the method modifies the matrix such that the new state
-   * corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two given faces
-   * at swapped positions. Of course, the two faces should not have a face/coface relation which each other ;
+   * corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but with the two given cells
+   * at swapped positions. Of course, the two cells should not have a face/coface relation which each other ;
    * \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    * 
-   * @param columnIndex1 @ref MatIdx index of the first face.
-   * @param columnIndex2 @ref MatIdx index of the second face. It is assumed that the @ref PosIdx of both only differs
+   * @param columnIndex1 @ref MatIdx index of the first cell.
+   * @param columnIndex2 @ref MatIdx index of the second cell. It is assumed that the @ref PosIdx of both only differs
    * by one if the barcode is maintained.
    * @return Let \f$ pos1 \f$ be the @ref PosIdx index of @p columnIndex1 and \f$ pos2 \f$ be the @ref PosIdx index of
    * @p columnIndex2. The method returns the @ref MatIdx of the column which has now, after the swap, the @ref PosIdx
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/columns/chain_column_extra_properties.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/columns/chain_column_extra_properties.h
index aca39824bf..5fb6d6acd9 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/columns/chain_column_extra_properties.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/columns/chain_column_extra_properties.h
@@ -66,14 +66,14 @@ class Chain_column_extra_properties
   /**
    * @brief Constructor setting the pivot at the given value and the pair to -1 (i.e. not paired).
    * 
-   * @param pivot @ref rowindex "Row index" of the pivot. Corresponds to the @ref IDIdx index of the face represented
+   * @param pivot @ref rowindex "Row index" of the pivot. Corresponds to the @ref IDIdx index of the cell represented
    * by the column.
    */
   Chain_column_extra_properties(ID_index pivot) : pivot_(pivot), pairedColumn_(-1) {}
   /**
    * @brief Constructor setting the pivot and the pair at the given values.
    * 
-   * @param pivot @ref rowindex "Row index" of the pivot. Corresponds to the @ref IDIdx index of the face represented
+   * @param pivot @ref rowindex "Row index" of the pivot. Corresponds to the @ref IDIdx index of the cell represented
    * by the column.
    * @param pair @ref MatIdx index of the pair of the column.
    */
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/matrix_dimension_holders.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/matrix_dimension_holders.h
index e743f989e8..30ce121a37 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/matrix_dimension_holders.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/matrix_dimension_holders.h
@@ -44,8 +44,8 @@ struct Dummy_matrix_dimension_holder
 /**
  * @ingroup persistence_matrix
  *
- * @brief Class managing the maximal dimension of a face represented in the inheriting matrix, when the option of
- * face removal is not enabled.
+ * @brief Class managing the maximal dimension of a cell represented in the inheriting matrix, when the option of
+ * cell removal is not enabled.
  * 
  * @tparam Dimension Dimension value type. Has to be an integer type.
  * If unsigned, the maximal value of the type should not be attained during a run.
@@ -75,7 +75,7 @@ class Matrix_max_dimension_holder
       : maxDim_(std::exchange(other.maxDim_, -1)){};
 
   /**
-   * @brief Returns the maximal dimension of a face represented in the matrix.
+   * @brief Returns the maximal dimension of a cell represented in the matrix.
    * 
    * @return The maximal dimension.
    */
@@ -106,8 +106,8 @@ class Matrix_max_dimension_holder
 /**
  * @ingroup persistence_matrix
  *
- * @brief Class managing the maximal dimension of a face represented in the inheriting matrix, when the option of
- * face removal is enabled.
+ * @brief Class managing the maximal dimension of a cell represented in the inheriting matrix, when the option of
+ * cell removal is enabled.
  * 
  * @tparam Dimension Dimension value type. Has to be an integer type.
  * If unsigned, the maximal value of the type should not be attained during a run.
@@ -141,7 +141,7 @@ class Matrix_all_dimension_holder
       : dimensions_(std::move(other.dimensions_)), maxDim_(std::exchange(other.maxDim_, -1)){};
 
   /**
-   * @brief Returns the maximal dimension of a face represented in the matrix.
+   * @brief Returns the maximal dimension of a cell represented in the matrix.
    * 
    * @return The maximal dimension.
    */
@@ -164,8 +164,8 @@ class Matrix_all_dimension_holder
   }
 
  protected:
-  std::vector<unsigned int> dimensions_;   /**< Number of faces by dimension. */
-  Dimension maxDim_;                  /**< Current maximal dimension. */
+  std::vector<unsigned int> dimensions_;  /**< Number of cells by dimension. */
+  Dimension maxDim_;                      /**< Current maximal dimension. */
 
   void update_up(unsigned int dimension) {
     if (dimensions_.size() <= dimension) dimensions_.resize(dimension + 1, 0);
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_pairing.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_pairing.h
index f44e518e78..e6afe9fc89 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_pairing.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_pairing.h
@@ -20,7 +20,7 @@
 
 #include <unordered_map>
 
-#include "boundary_face_position_to_id_mapper.h"
+#include "boundary_cell_position_to_id_mapper.h"
 
 namespace Gudhi {
 namespace persistence_matrix {
@@ -48,7 +48,7 @@ struct Dummy_ru_pairing
 template <class Master_matrix>
 class RU_pairing : public std::conditional<
                        Master_matrix::Option_list::has_removable_columns,
-                       Face_position_to_ID_mapper<typename Master_matrix::ID_index, typename Master_matrix::Pos_index>,
+                       Cell_position_to_ID_mapper<typename Master_matrix::ID_index, typename Master_matrix::Pos_index>,
                        Dummy_pos_mapper
                     >::type
 {
@@ -57,9 +57,9 @@ class RU_pairing : public std::conditional<
   using ID_index = typename Master_matrix::ID_index;
   //PIDM = Position to ID Map
   using PIDM = typename std::conditional<Master_matrix::Option_list::has_removable_columns,
-                                        Face_position_to_ID_mapper<ID_index, Pos_index>,
-                                        Dummy_pos_mapper
-                                       >::type;
+                                          Cell_position_to_ID_mapper<ID_index, Pos_index>,
+                                          Dummy_pos_mapper
+                                        >::type;
 
  public:
   using Barcode = typename Master_matrix::Barcode;  /**< Barcode type. */
@@ -93,7 +93,7 @@ class RU_pairing : public std::conditional<
   Barcode barcode_;       /**< Bar container. */
   Dictionary indexToBar_; /**< Map from @ref MatIdx index to bar index. */
   /**
-   * @brief Map from face ID to face position. Only stores a pair if ID != position.
+   * @brief Map from cell ID to cell position. Only stores a pair if ID != position.
    */
   std::unordered_map<ID_index, Pos_index> idToPosition_;  //TODO: test other map types
 
diff --git a/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_vine_swap.h b/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_vine_swap.h
index 64561c8683..00c76749f7 100644
--- a/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_vine_swap.h
+++ b/src/Persistence_matrix/include/gudhi/Persistence_matrix/ru_vine_swap.h
@@ -25,7 +25,7 @@
 #include <stdexcept>    //std::invalid_argument
 
 #include "ru_pairing.h"
-#include "boundary_face_position_to_id_mapper.h"
+#include "boundary_cell_position_to_id_mapper.h"
 
 namespace Gudhi {
 namespace persistence_matrix {
@@ -68,7 +68,7 @@ class RU_vine_swap : public std::conditional<Master_matrix::Option_list::has_col
                      public std::conditional<Master_matrix::Option_list::has_column_pairings &&
                                                 Master_matrix::Option_list::has_removable_columns, 
                                              Dummy_pos_mapper,
-                                             Face_position_to_ID_mapper<typename Master_matrix::ID_index,
+                                             Cell_position_to_ID_mapper<typename Master_matrix::ID_index,
                                                                         typename Master_matrix::Pos_index>
                                             >::type
 {
@@ -98,20 +98,20 @@ class RU_vine_swap : public std::conditional<Master_matrix::Option_list::has_col
    * @brief Does the same than @ref vine_swap, but assumes that the swap is non trivial and
    * therefore skips a part of the case study.
    * 
-   * @param index @ref PosIdx index of the first face to swap. The second one has to be at `position + 1`.
+   * @param index @ref PosIdx index of the first cell to swap. The second one has to be at `position + 1`.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
    */
   bool vine_swap_with_z_eq_1_case(Pos_index index);
   /**
-   * @brief Does a vine swap between two faces which are consecutive in the filtration.
+   * @brief Does a vine swap between two cells which are consecutive in the filtration.
    * Roughly, if \f$ F \f$ is the current filtration represented by the matrix, the method modifies the matrix
    * such that the new state corresponds to a valid state for the filtration \f$ F' \f$ equal to \f$ F \f$ but
-   * with the two faces at position `position` and `position + 1` swapped. Of course, the two faces should
+   * with the two cells at position `position` and `position + 1` swapped. Of course, the two cells should
    * not have a face/coface relation which each other ; \f$ F' \f$ has to be a valid filtration.
    * See @cite vineyards for more information about vine and vineyards.
    * 
-   * @param index @ref PosIdx index of the first face to swap. The second one has to be at `position + 1`.
+   * @param index @ref PosIdx index of the first cell to swap. The second one has to be at `position + 1`.
    * @return true If the barcode changed from the swap.
    * @return false Otherwise.
    */
@@ -129,8 +129,8 @@ class RU_vine_swap : public std::conditional<Master_matrix::Option_list::has_col
       swap(static_cast<RU_pairing<Master_matrix>&>(swap1), static_cast<RU_pairing<Master_matrix>&>(swap2));
     }
     if (!Master_matrix::Option_list::has_column_pairings || !Master_matrix::Option_list::has_removable_columns) {
-      swap(static_cast<Face_position_to_ID_mapper<ID_index, Pos_index>&>(swap1),
-           static_cast<Face_position_to_ID_mapper<ID_index, Pos_index>&>(swap2));
+      swap(static_cast<Cell_position_to_ID_mapper<ID_index, Pos_index>&>(swap1),
+           static_cast<Cell_position_to_ID_mapper<ID_index, Pos_index>&>(swap2));
     }
   }
 
@@ -144,7 +144,7 @@ class RU_vine_swap : public std::conditional<Master_matrix::Option_list::has_col
   using RUM = typename std::conditional<Master_matrix::Option_list::has_column_pairings &&
                                             Master_matrix::Option_list::has_removable_columns,
                                         Dummy_pos_mapper,
-                                        Face_position_to_ID_mapper<ID_index, Pos_index>
+                                        Cell_position_to_ID_mapper<ID_index, Pos_index>
                                        >::type;
   constexpr auto& _positionToRowIdx();
 
diff --git a/src/Persistence_matrix/test/pm_matrix_tests.h b/src/Persistence_matrix/test/pm_matrix_tests.h
index c6785b9b92..dd20612656 100644
--- a/src/Persistence_matrix/test/pm_matrix_tests.h
+++ b/src/Persistence_matrix/test/pm_matrix_tests.h
@@ -902,7 +902,7 @@ void test_ru_maximal_simplex_removal() {
   }
 
   if constexpr (Matrix::Option_list::has_vine_update) {
-    m.remove_maximal_face(6);
+    m.remove_maximal_cell(6);
   } else {
     m.remove_last();
   }
@@ -927,7 +927,7 @@ void test_chain_maximal_simplex_removal(Matrix& m) {
 
   if constexpr (Matrix::Option_list::has_vine_update && Matrix::Option_list::has_map_column_container &&
                 Matrix::Option_list::has_column_pairings) {
-    m.remove_maximal_face(6);
+    m.remove_maximal_cell(6);
   } else {
     m.remove_last();
   }
@@ -957,7 +957,7 @@ void test_maximal_dimension(Matrix& m) {
   if constexpr (Matrix::Option_list::has_vine_update &&
                 (Matrix::Option_list::is_of_boundary_type ||
                  (Matrix::Option_list::has_map_column_container && Matrix::Option_list::has_column_pairings))) {
-    m.remove_maximal_face(7);
+    m.remove_maximal_cell(7);
     BOOST_CHECK_EQUAL(m.get_max_dimension(), 3);
   }
 }

From b4694d930d810c26c5aa5234dc059ce2f771595f Mon Sep 17 00:00:00 2001
From: Vincent Rouvreau <vincent.rouvreau@inria.fr>
Date: Thu, 26 Sep 2024 15:59:43 +0200
Subject: [PATCH 5/6] code review: just test if windows for pykeops

---
 .../modules/GUDHI_third_party_libraries.cmake      | 14 ++++----------
 1 file changed, 4 insertions(+), 10 deletions(-)

diff --git a/src/cmake/modules/GUDHI_third_party_libraries.cmake b/src/cmake/modules/GUDHI_third_party_libraries.cmake
index ffed360461..a72d106bb2 100644
--- a/src/cmake/modules/GUDHI_third_party_libraries.cmake
+++ b/src/cmake/modules/GUDHI_third_party_libraries.cmake
@@ -186,18 +186,12 @@ if (WITH_GUDHI_PYTHON)
     find_python_module("pydata_sphinx_theme")
     find_python_module("sphinxcontrib.bibtex")
     find_python_module("networkx")
+    # Specific case for PyKeops that can be imported on Windows, but fails because it uses fcntl (not available on Windows)
+    if (NOT WIN32)
+      find_python_module("pykeops")
+    endif()
   endif()
 
-  # Specific case for PyKeops that can be imported on Windows, but fails because it uses fcntl (not available on Windows)
-  # Also fcntl has no metadata, so find_python_module does not work
-  # "import fcntl" is about 1 sec. faster than "import pykeops"
-  execute_process(
-          COMMAND ${Python_EXECUTABLE}  -c "import fcntl"
-          RESULT_VARIABLE FCNTL_IMPORT_MODULE_RESULT
-          OUTPUT_VARIABLE FCNTL_IMPORT_MODULE_OUPUT)
-  if(FCNTL_IMPORT_MODULE_RESULT EQUAL 0)
-    find_python_module("pykeops")
-  endif()
 
   if(NOT GUDHI_PYTHON_PATH)
     message(FATAL_ERROR "ERROR: GUDHI_PYTHON_PATH is not valid.")

From 9fbea63b0db06a40a80d5be379bbfda24677985a Mon Sep 17 00:00:00 2001
From: Vincent Rouvreau <vincent.rouvreau@inria.fr>
Date: Thu, 26 Sep 2024 16:51:05 +0200
Subject: [PATCH 6/6] [skip ci] code review: Rework comment. Remove comments
 that were no more relevant.

---
 src/cmake/modules/GUDHI_third_party_libraries.cmake | 2 +-
 src/python/CMakeLists.txt                           | 2 --
 2 files changed, 1 insertion(+), 3 deletions(-)

diff --git a/src/cmake/modules/GUDHI_third_party_libraries.cmake b/src/cmake/modules/GUDHI_third_party_libraries.cmake
index a72d106bb2..6527d0be3c 100644
--- a/src/cmake/modules/GUDHI_third_party_libraries.cmake
+++ b/src/cmake/modules/GUDHI_third_party_libraries.cmake
@@ -186,7 +186,7 @@ if (WITH_GUDHI_PYTHON)
     find_python_module("pydata_sphinx_theme")
     find_python_module("sphinxcontrib.bibtex")
     find_python_module("networkx")
-    # Specific case for PyKeops that can be imported on Windows, but fails because it uses fcntl (not available on Windows)
+    # Specific case for PyKeops on Windows where import fails because it uses fcntl (not available on Windows)
     if (NOT WIN32)
       find_python_module("pykeops")
     endif()
diff --git a/src/python/CMakeLists.txt b/src/python/CMakeLists.txt
index e68e8f2c64..3198461820 100644
--- a/src/python/CMakeLists.txt
+++ b/src/python/CMakeLists.txt
@@ -142,7 +142,6 @@ else()
   disable_python_documentation("ot")
 endif()
 if(HNSWLIB_FOUND)
-  # Does not have a version number...
   add_gudhi_debug_info("HNSWlib version ${HNSWLIB_VERSION}")
 endif()
 if(TORCH_FOUND)
@@ -175,7 +174,6 @@ else()
   disable_python_documentation("pydata_sphinx_theme")
 endif()
 if(SPHINXCONTRIB.BIBTEX_FOUND)
-  # Does not have a version number...
   add_gudhi_debug_info("sphinxcontrib-bibtex version ${SPHINXCONTRIB.BIBTEX_VERSION}")
 else()
   disable_python_documentation("sphinxcontrib-bibtex")