Homotopy hypothesis

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces.

One version of the hypothesis was claimed to be proved in the 1991 paper by Kapranov and Voevodsky. Their proof turned out to be flawed and their result in the form interpreted by Carlos Simpson is now known as the Simpson conjecture.

Formulations
There are many ways to formulate the hypothesis. For example, if we model our ∞-groupoids as Kan complexes (quasi-categories ), then the homotopy types of the geometric realizations of these sets give models for every homotopy type (perhaps in the weak form). It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types.

Depending on the definitions of ∞-groupoids, the hypothesis may trivially hold.