Simplicial homotopy

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

[math]\displaystyle{ f, g: X \to Y }[/math]

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

[math]\displaystyle{ h: X \times \Delta^{1} \to Y }[/math]

such that the diagram (see [1]) formed by f, g and h commute; the key is to use the diagram that results in [math]\displaystyle{ f(x) = h(x, 0) }[/math] and [math]\displaystyle{ g(x) = h(x, 1) }[/math] for all x in X.

See also

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

External links

• http://ncatlab.org/nlab/show/simplicial+homotopy