# Cell complexes and their homology¶

Sage includes some tools for algebraic topology: from the algebraic side, chain complexes and their homology, and from the topological side, simplicial complexes, \(\Delta\)-complexes, cubical complexes, and simplicial sets. A class of generic cell complexes is also available, mainly for developers who want to use it as a base for other types of cell complexes.

- Chain complexes
- Chains and cochains
- Morphisms of chain complexes
- Chain homotopies and chain contractions
- Homspaces between chain complexes
- Finite simplicial complexes
- Morphisms of simplicial complexes
- Homsets between simplicial complexes
- Examples of simplicial complexes
- Finite Delta-complexes
- Finite cubical complexes
- Simplicial sets
- Methods of constructing simplicial sets
- Examples of simplicial sets.
- Catalog of simplicial sets
- Morphisms and homsets for simplicial sets
- Generic cell complexes
- Koszul Complexes
- Hochschild Complexes
- Homology Groups
- Homology and cohomology with a basis
- Algebraic topological model for a cell complex
- Induced morphisms on homology
- Utility Functions for Matrices
- Interface to CHomP