AbstractSimplicialComplexes -- a package for working with abstract simplicial complexes

Description

In this package our conventions are that `abstract simplicial complexes' have vertices supported on the set [n] := {1,...,n}. Our aim is to provide a methology for working with such objects directly. In this regard our approach differs from that of the package SimplicialComplexes.m2. Here, we are especially interested in homological aspects of SimplicialComplexes 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

How to make abstract simplicial complexes -- Using the type AbstractSimplicialComplexs to represent abstract simplicial complexes
How to make reduced and non-reduced simplicial chain complexes -- Simplicial homological constructors
Calculations with random simplicial complexes -- Homological calculations on random simplicial complexes

Author

Nathan Grieve <[email protected]>

Version

This documentation describes version 0.1 of AbstractSimplicialComplexes.

Source code

The source code from which this documentation is derived is in the file AbstractSimplicialComplexes.m2.

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 -- Induced maps that arise via inclusions of abstract simplicial complexes
- inducedSimplicialChainComplexMap -- Induced maps that arise via inclusions of abstract simplicial complexes
- randomAbstractSimplicialComplex -- Create a random simplicial set
- randomSubSimplicialComplex -- Create a random sub-simplicial 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 -- Induced maps that arise via inclusions of abstract simplicial complexes
- inducedSimplicialChainComplexMap(AbstractSimplicialComplex,AbstractSimplicialComplex) -- see inducedSimplicialChainComplexMap -- Induced maps that arise via inclusions of abstract simplicial complexes
- new AbstractSimplicialComplex
- randomAbstractSimplicialComplex(ZZ) -- see randomAbstractSimplicialComplex -- Create a random simplicial set
- randomAbstractSimplicialComplex(ZZ,ZZ) -- see randomAbstractSimplicialComplex -- Create a random simplicial set
- randomAbstractSimplicialComplex(ZZ,ZZ,ZZ) -- see randomAbstractSimplicialComplex -- Create a random simplicial set
- randomSubSimplicialComplex(AbstractSimplicialComplex) -- see randomSubSimplicialComplex -- Create a random sub-simplicial 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.