# Abstract Simplicial Complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a simplicial complex, consisting of a family of finite sets closed under the operation of taking subsets. In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems.

Read more about Abstract Simplicial ComplexDefinitions, Geometric Realization, Examples, Enumeration

### Other articles related to "complex, simplicial, abstract simplicial complex, simplicial complex":

The Definition of The Cotangent Complex
... The correct definition of the cotangent complex begins in the homotopical setting ... Quillen and André worked with the simplicial commutative rings, while Illusie worked with simplicial ringed topoi ... For simplicity, we will consider only the case of simplicial commutative rings ...
Simplicial Polytope
... In geometry, a simplicial polytope is a d-polytope whose facets are all simplices ... For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph ... Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons ...
Acyclic Model - Example
... Let be the singular chain complex functor and be the simplicial chain complex functor ... and -acyclic (the proof that is not entirely straigtforward and uses a detour through simplicial subdivision, which can also be handled using the above theorem) ... It is rather obvious that and so we conclude that singular and simplicial homology are isomorphic on ...
Abstract Simplicial Complex - Enumeration
... The number of abstract simplicial complexes on n elements is the nth Dedekind number ... These numbers grow very rapidly, and are known only for n ≤ 8 they are 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 in OEIS) ...
Family Of Sets - Related Concepts
... An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line segments, triangles ... In an abstract simplicial complex, each simplex is represented simply as the set of its vertices ... also belong to the family forms an abstract simplicial complex ...

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