In algebraic topology, a simplicial homotopy^{[1]pg 23} is an analog of a homotopy between topological spaces for simplicial sets. 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 diagram (see ) formed by f, g and h commute; the key is to use the diagram that results in ${\displaystyle f(x)=h(x,0)}$ and ${\displaystyle g(x)=h(x,1)}$ for all x in X.

• Kan complex
• Dold–Kan correspondence (under which a chain homotopy corresponds to a simplicial homotopy)
• Simplicial homology