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In algebraic topology, a simplicial homotopypg 23 is an analog of a
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
between topological spaces for
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s. If :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ \to Y such that the diagram (se

formed by ''f'', ''g'' and ''h'' commute; the key is to use the diagram that results in f(x) = h(x, 0) and g(x) = h(x, 1) for all ''x'' in ''X''.


See also

*
Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
*
Dold–Kan correspondence In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an Equivalence of categories, equivalence between the category of (nonnegatively graded) ...
(under which a
chain homotopy In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain ...
corresponds to a simplicial homotopy) *
Simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...


References


External links

*http://ncatlab.org/nlab/show/simplicial+homotopy Homotopy theory Simplicial sets {{topology-stub