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In algebraic topology, a simplicial homotopy [1] pg 23 is an analog of a homotopy between topological spaces for simplicial sets. If ,: are maps between simplicial sets, a simplicial homotopy from f to g is a map :
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. [2]: sec.8.6 As a result, it gives a computable way to distinguish one space from another.
The same homotopy category can arise from many different model categories. An important example is the standard model structure on simplicial sets: the associated homotopy category is equivalent to the homotopy category of topological spaces, even though simplicial sets are combinatorially defined objects that lack any topology.
between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them. It is part of the general definition of a Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations ...
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. ... Each simplicial set = ...
Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of homotopy groups of finite simplicial complexes. Gmsh includes a homology solver for finite element meshes, which can generate Cohomology bases directly usable by finite element software. [14]
The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees with the homotopy groups of the topological space which realizes it.
A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex . In fact it can be shown that any simplicial abelian group A {\displaystyle A} is non-canonically homotopy equivalent to a product of Eilenberg–MacLane ...