Simplicial homotopy

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,^{pg 23} if

${\displaystyle f,g:X\to Y}$

are maps between simplicial sets, a simplicial homotopy from f to g is a map

${\displaystyle h:X\times \Delta ^{1}\to Y}$

such that the restriction of ${\displaystyle h}$ along ${\displaystyle X\simeq X\times \Delta ^{0}{\overset {0}{\hookrightarrow }}X\times \Delta ^{1}}$ is ${\displaystyle f}$ and the restriction along ${\displaystyle 1}$ is ${\displaystyle g}$; see . In particular, ${\displaystyle f(x)=h(x,0)}$ and ${\displaystyle g(x)=h(x,1)}$ for all x in X.

Using the adjunction

${\displaystyle \operatorname {Hom} (X\times \Delta ^{1},Y)=\operatorname {Hom} (\Delta ^{1}\times X,Y)=\operatorname {Hom} (\Delta ^{1},{\underline {\operatorname {Hom} }}(X,Y))}$,

the simplicial homotopy ${\displaystyle h}$ can also be thought of as a path in the simplicial set ${\displaystyle {\underline {\operatorname {Hom} }}(X,Y).}$

A simplicial homotopy is in general not an equivalence relation. However, if ${\displaystyle {\underline {\operatorname {Hom} }}(X,Y)}$ is a Kan complex (e.g., if ${\displaystyle Y}$ is a Kan complex), then a homotopy from ${\displaystyle f:X\to Y}$ to ${\displaystyle g:X\to Y}$ is an equivalence relation. Indeed, a Kan complex is an ∞-groupoid; i.e., every morphism (path) is invertible. Thus, if h is a homotopy from f to g, then the inverse of h is a homotopy from g to f, establishing that the relation is symmetric. The transitivity holds since a composition is possible.