Kan fibration

In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.

For each n ≥ 0, recall that the standard ${\displaystyle n}$-simplex, ${\displaystyle \Delta ^{n}}$, is the representable simplicial set

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard ${\displaystyle n}$-simplex: the convex subspace of ℝⁿ⁺¹ consisting of all points ${\displaystyle (t_{0},\dots ,t_{n})}$ such that the coordinates are non-negative and sum to 1.

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