∞-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 (with the standard model structure).[1] 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
Grothendieck suggested in Pursuing Stacks that there should be an extraordinarily simple model of ∞-groupoids using globular sets. These 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.
See also
References
External links
- infinity-groupoid in nLab
- Maltsiniotis, Georges, Grothendieck ∞-groupoids,and still another definition of ∞-categories (PDF)
- Zawadowski, Marek, Introduction to Test Categories (PDF)