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In algebraic topology, a simplicial homotopy 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.

See also

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

References

1. Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 978-3-0346-0188-7. OCLC 837507571.

External links

• simplicial homotopy

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