∞-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 of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.
The homotopy hypothesis states that ∞-groupoids are spaces.

Globular Groupoids

suggested in Pursuing Stacks that there should be an extraordinarily simple model of ∞-groupoids using globular sets. These sets are constructed as presheaves on the globular category. This is defined as the category whose objects are finite ordinals and morphisms are given by
such that the globular relations hold
These encode the fact that -morphisms should not be able to see -morphisms. We can also consider globular objects in a category as functors
There was hope originally that such a strict model would be sufficient for homotopy theory, but there is evidence suggesting otherwise.