∞-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

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

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

such that the globular relations hold

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

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

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

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

References

1. "Kan complex in nLab".

External links

• infinity-groupoid in 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