Kan fibration

In algebraic topology, a branch of mathematics, Kan complexes are a tool for combinatorial definition of homotopy groups.

Definition

A simplicial set is a Kan complex if it satisfies the Kan extension property:

Homotopy

DM Kan gave a combinatorial definition of homotopy groups for Kan complexes.

For example, the singular chain complex

Be a topological space and its singular chain complex: the n- simplices of are according to the continuous maps of the standard n -simplex.

Is a Kan complex, its homotopy groups (in the sense of Kan) agree with the homotopy groups of the same.