Milnor's theorem on Kan complexes

In mathematics, especially algebraic topology, a theorem of Milnor says that the geometric realization functor from the homotopy category of the category Kan of Kan complexes to the homotopy category of the category Top of (reasonable) topological spaces is fully faithful. The theorem in particular implies Kan and Top have the same homotopy category.

In today’s language, Kan is typically identified as ∞-Grpd, the category of ∞-groupoids. Thus, the theorem can be viewed as an instance of Grothendieck's homotopy hypothesis which says ∞-groupoids are spaces (or that they can model spaces from the homotopy theory point of view).

The pointed version of the theorem is also true.