Kan complex

Summary

A Kan complex is a simplicial set in which every horn can be filled to a full simplex. Kan complexes behave like topological spaces. Their simplicial homotopy groups are well defined and agree with those of their geometric realization. The singular simplicial set of every space is a Kan complex. Kan complexes are the fibrant objects of the Kan–Quillen model structure on simplicial sets. They are a standard model of $\infty$-groupoids.