Simplicial homotopy explained

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 \to Y are maps between simplicial sets, a simplicial homotopy from f to g is a maph: X x \Delta ^{1} \to Y such that the diagram (see https://books.google.com/books?id=iFv2BwAAQBAJ&dq=%22simplicial+homotopy%22&pg=PA23) formed by f, g and h commute; the key is to use the diagram that results in f(x) = h(x, 0 ) and 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

External links

• simplicial homotopy

Notes and References

1. Book: Goerss. Paul G.. Simplicial Homotopy Theory. Jardin. John F.. 2009. Birkhäuser Basel. 978-3-0346-0188-7. 837507571.