AbstractSimplicialComplexes -- a package for working with abstract simplicial complexes

Description

The package AbstractSimplicialComplexes provides a methodology for working with abstract simplicial complexes. The starting point is to represent each abstract simplicial complex as a graded list. In this regard our approach differs fundamentally from that of the package SimplicialComplexes.

In this package our conventions are that abstract simplicial complexes have vertices supported on the set $[n] := \{1,\dots,n\}$. Users who wish to use the package to study simplicial complexes on vertex sets different from $[n]$ should first fix a suitable order preserving bijection which is compatible with the standard lexicographic ordering.

Here, we are especially interested in homological aspects of abstract simplicial complexes and our approach is to implement such simplicial complexes as certain graded lists. In particular, we provide methods for working with the chain complexes that are associated to each abstract simplicial complex. We also give some functionality for producing random simplicial complexes.

An overview of this package

For an overview of how to use this package, see

Mathematical background, motivation, conventions and notations

Our conventions for abstract simplicial complexes, their associated boundary maps, reduced and non-reduced chain complexes are similar to those of the texts

Combinatorial commutative algebra by E. Miller and B. Sturmfels [Springer-Verlag, (2005)]
Basic Topology by M. A. Armstrong [Springer-Verlag (1983)]
Cohen-Macaulay rings by W. Bruns and J. Herzog [Cambridge University Press (1993)]

and others.

This package also has strong motivation from and interplay with Topological Data Analysis. We refer to the works

Topology and data by G. Carlsson [Bull. Amer. Math. Soc. vol. 46, no. 2 (2009)]
Topological data analysis with applications by G. Carlsson and M. Vejdemo-Johansson [Cambridge University Press (2022)]

and the references therein for further details about abstract simplicial complexes within the context of Topological Data Analysis.

This package also provides constructors for producing certain classes of random simplicial complexes. We refer to the work

Cohen-Lenstra heuristics for torsion in homology of random complexes by M. Kahle, F. H. Lutz, A. Newman, and K. Parsons [Exp. Math. vol. 29, no. 3 (2020)]

and the references therein for further details about constructing random simplicial complexes.

Author

Nathan Grieve <[email protected]>

Certification

Version 1.0 of this package was accepted for publication in volume 15 of Journal of Software for Algebra and Geometry on 2025-04-14, in the article Abstract simplicial complexes in Macaulay2 (DOI: 10.2140/jsag.2025.15.29). That version can be obtained from the journal.

Version

This documentation describes version 1.1 of AbstractSimplicialComplexes.

Citation

If you have used this package in your research, please cite it as follows:

@misc{AbstractSimplicialComplexesSource,
  title = {{AbstractSimplicialComplexes: A \emph{Macaulay2} package. Version~1.1}},
  author = {Nathan Grieve},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

@article{AbstractSimplicialComplexesArticle,
  title = {{Abstract simplicial complexes in \emph{Macaulay2}}},
  author = {Nathan Grieve},
  journal = {Journal of Software for Algebra and Geometry},
  volume = {15},
  year = {2025},
}

Exports

Types
- AbstractSimplicialComplex -- the type of all abstract simplicial complexes
Functions and commands
- abstractSimplicialComplex -- the AbstractSimplicialComplex that is determined by an abstract simplicial complex
- abstractSimplicialComplexFacets -- the facets of a simplicial complex
- ambientAbstractSimplicialComplex -- the ambient simplex
- ambientAbstractSimplicialComplexSize -- the ambient simplex size
- inducedReducedSimplicialChainComplexMap -- the induced maps that arise via inclusions of abstract simplicial complexes
- inducedSimplicialChainComplexMap -- the induced maps that arise via inclusions of abstract simplicial complexes
- randomAbstractSimplicialComplex -- create a random abstract simplicial complex
- randomSubSimplicialComplex -- create a random subsimplicial complex
- reducedSimplicialChainComplex -- the reduced homological chain complex that is determined by an abstract simplicial complex
- simplicialChainComplex -- the non-reduced homological chain complex that is determined by an abstract simplicial complex
Methods
- abstractSimplicialComplex(List) -- see abstractSimplicialComplex -- the AbstractSimplicialComplex that is determined by an abstract simplicial complex
- abstractSimplicialComplex(ZZ) -- see abstractSimplicialComplex -- the AbstractSimplicialComplex that is determined by an abstract simplicial complex
- abstractSimplicialComplex(ZZ,ZZ) -- see abstractSimplicialComplex -- the AbstractSimplicialComplex that is determined by an abstract simplicial complex
- AbstractSimplicialComplex == AbstractSimplicialComplex -- decide if two simplicial complexes are equal
- AbstractSimplicialComplex _ ZZ -- the $k$ faces of a simplicial complex
- abstractSimplicialComplexFacets(AbstractSimplicialComplex) -- see abstractSimplicialComplexFacets -- the facets of a simplicial complex
- ambientAbstractSimplicialComplex(AbstractSimplicialComplex) -- see ambientAbstractSimplicialComplex -- the ambient simplex
- ambientAbstractSimplicialComplexSize(AbstractSimplicialComplex) -- see ambientAbstractSimplicialComplexSize -- the ambient simplex size
- describe(AbstractSimplicialComplex) -- real description
- dim(AbstractSimplicialComplex) -- the dimension of a simplicial complex
- inducedReducedSimplicialChainComplexMap(AbstractSimplicialComplex,AbstractSimplicialComplex) -- see inducedReducedSimplicialChainComplexMap -- the induced maps that arise via inclusions of abstract simplicial complexes
- inducedSimplicialChainComplexMap(AbstractSimplicialComplex,AbstractSimplicialComplex) -- see inducedSimplicialChainComplexMap -- the induced maps that arise via inclusions of abstract simplicial complexes
- new AbstractSimplicialComplex
- randomAbstractSimplicialComplex(ZZ) -- see randomAbstractSimplicialComplex -- create a random abstract simplicial complex
- randomAbstractSimplicialComplex(ZZ,ZZ) -- see randomAbstractSimplicialComplex -- create a random abstract simplicial complex
- randomAbstractSimplicialComplex(ZZ,ZZ,ZZ) -- see randomAbstractSimplicialComplex -- create a random abstract simplicial complex
- randomSubSimplicialComplex(AbstractSimplicialComplex) -- see randomSubSimplicialComplex -- create a random subsimplicial complex
- reducedSimplicialChainComplex(AbstractSimplicialComplex) -- see reducedSimplicialChainComplex -- the reduced homological chain complex that is determined by an abstract simplicial complex
- simplicialChainComplex(AbstractSimplicialComplex) -- see simplicialChainComplex -- the non-reduced homological chain complex that is determined by an abstract simplicial complex

For the programmer

The object AbstractSimplicialComplexes is a package, defined in AbstractSimplicialComplexes.m2.

The source of this document is in AbstractSimplicialComplexes.m2:454:0.