en.wikipedia.org/wiki/Homotopy_hypothesis
1 correction found
This version is still open.
This is overstated: the n-groupoid homotopy hypothesis has already been proved for some standard weak n-groupoid models, such as Tamsamani's.
Full reasoning
The sentence is inaccurate because it states, without qualification, that the n-version of the homotopy hypothesis is still open.
But for at least one standard notion of weak n-groupoid, the hypothesis was proved long ago. In his 1996 paper, Zouhair Tamsamani states that geometric realization and the Poincaré n-groupoid functor induce an equivalence between the category of n-groupoids and the category of n-truncated topological spaces, after localizing at weak equivalences. That is precisely a proof of the homotopy hypothesis for Tamsamani weak n-groupoids.
Later work also treats other algebraic models of n-types and compares them directly with Tamsamani weak n-groupoids. For example, Blanc and Paoli (2014) write that their weakly globular n-fold groupoids model n-types and compare these models to Tamsamani's weak n-groupoids.
So the unqualified sentence "This version is still open." is false as written. What is true is that the status depends on which definition of weak n-groupoid one chooses; some formulations remain open, while others have been proved.
2 sources
- Equivalence de la théorie homotopique des n-groupoïdes et celle des espaces topologiques n-tronqués
Abstract: This paper ... show[s] that the functors geometric realisation and Poincaré n-groupoid induce an equivalence between the category of n-groupoids and the category of n-truncated topological spaces, when we localise both categories by weak equivalence.
- Segal-type algebraic models of n-types
Abstract: For each n≥1 we introduce two new Segal-type models of n-types ... We show that any n-type can be represented up to homotopy by such models ... We compare these models to Tamsamani's weak n-groupoids.