In mathematics, especially

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, a theorem of Milnor says that the geometric realization functor from the

homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed ...

of the category Kan of

Kan complex 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 structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...

es to the homotopy category of the category Top of (reasonable) topological spaces is

fully faithful In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a fully faithful functor. Formal definitions Explicitly, let ''C'' and ' ...

. 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

∞-groupoid In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standa ...

s. Thus, the theorem can be viewed as an instance of Grothendieck's

homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by M ...

which says ∞-groupoids are spaces (or that they can model spaces from the

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

point of view). The pointed version of the theorem is also true.

Proof

A key step in the proof of the theorem is the following result (which is also sometimes called Milnor's theorem): Indeed, the above says that

\eta : \operatorname \to \operatorname(,  -, )

is invertible on the homotopy category or, equivalently,

,  - ,

is fully faithful there.

References

Sources

* * {{topology-stub Theorems in homotopy theory