Simplicial homotopy

In algebraic topology, a simplicial homotopy[1]pg 23 is an analog of a homotopy between topological spaces for simplicial sets. If

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

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

h : X × Δ 1 → Y {\displaystyle h:X\times \Delta ^{1}\to Y}

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

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


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