∞-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). 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

Alexander Grothendieck 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 $\mathbb {G}$. This is defined as the category whose objects are finite ordinals $[n]$ and morphisms are given by

${\begin{aligned}\sigma _{n}:[n]\to [n+1]\\\tau _{n}:[n]\to [n+1]\end{aligned}}$

such that the globular relations hold

${\begin{aligned}\sigma _{n+1}\circ \sigma _{n}&=\tau _{n+1}\circ \sigma _{n}\\\sigma _{n+1}\circ \tau _{n}&=\tau _{n+1}\circ \tau _{n}\end{aligned}}$

These encode the fact that $n$-morphisms should not be able to see $(n+1)$-morphisms. We can also consider globular objects in a category ${\mathcal {C}}$ as functors

$X_{\bullet }\colon \mathbb {G} ^{op}\to {\mathcal {C}}.$

There was hope originally that such a strict model would be sufficient for homotopy theory, but there is evidence suggesting otherwise.

See also

• Groupoid
• Homotopy type theory

External links

• infinity-groupoid at the nLab
• Maltsiniotis, Georges (2010), "Grothendieck ∞-groupoids, and still another definition of ∞-categories", arXiv:1009.2331
• Zawadowski, Marek, Introduction to Test Categories (PDF), archived from the original (PDF) on 2015-03-26