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From Wikipedia, the free encyclopedia

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.[2]: 2–3 [3]

Globular Groupoids

Alexander Grothendieck suggested in Pursuing Stacks[2]: 3–4, 201  that there should be an extraordinarily simple model of ∞-groupoids using globular sets, originally called hemispherical complexes. 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. When writing these down as a globular set , the source and target maps are then written as

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. It turns out for its associated homotopy -type can never be modeled as a strict globular groupoid for .[2]: 445 [4] This is because strict ∞-groupoids only model spaces with a trivial Whitehead product.[5]

Examples

Fundamental ∞-groupoid

Given a topological space there should be an associated fundamental ∞-groupoid where the objects are points 1-morphisms are represented as paths, 2-morphisms are homotopies of paths, 3-morphisms are homotopies of homotopies, and so on. From this infinity groupoid we can find an -groupoid called the fundamental -groupoid whose homotopy type is that of .

Note that taking the fundamental ∞-groupoid of a space such that is equivalent to the fundamental n-groupoid . Such a space can be found using the Whitehead tower.

Abelian globular groupoids

One useful case of globular groupoids comes from a chain complex which is bounded above, hence let's consider a chain complex .[6] There is an associated globular groupoid. Intuitively, the objects are the elements in , morphisms come from through the chain complex map , and higher -morphisms can be found from the higher chain complex maps . We can form a globular set with

and the source morphism is the projection map

and the target morphism is the addition of the chain complex map together with the projection map. This forms a globular groupoid giving a wide class of examples of strict globular groupoids. Moreover, because strict groupoids embed inside weak groupoids, they can act as weak groupoids as well.

Applications

Higher local systems

One of the basic theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid to the category of Abelian groups, the category of -modules, or some other abelian category. That is, a local system is equivalent to giving a functor

generalizing such a definition requires us to consider not only an abelian category, but also its derived category. A higher local system is then an ∞-functor

with values in some derived category. This has the advantage of letting the higher homotopy groups to act on the higher local system, from a series of truncations. A toy example to study comes from the Eilenberg–MacLane spaces , or by looking at the terms from the Whitehead tower of a space. Ideally, there should be some way to recover the categories of functors from their truncations and the maps whose fibers should be the categories of -functors

Another advantage of this formalism is it allows for constructing higher forms of -adic representations by using the etale homotopy type of a scheme and construct higher representations of this space, since they are given by functors

Higher gerbes

Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes. Over a space an n-gerbe should be an object such that when restricted to a small enough subset , is represented by an n-groupoid, and on overlaps there is an agreement up to some weak equivalence. Assuming the homotopy hypothesis is correct, this is equivalent to constructing an object such that over any open subset

is an n-group, or a homotopy n-type. Because the nerve of a category can be used to construct an arbitrary homotopy type, a functor over an site , e.g.

will give an example of a higher gerbe if the category lying over any point is a non-empty category. In addition, it would be expected this category would satisfy some sort of descent condition.

See also

References

  1. ^ "Kan complex in nLab".
  2. ^ a b c Grothendieck. "Pursuing Stacks". thescrivener.github.io. Archived (PDF) from the original on 30 Jul 2020. Retrieved 2020-09-17.
  3. ^ Maltsiniotis, Georges. "Grothendieck infinity groupoids and still another definition of infinity categories" (PDF). Archived (PDF) from the original on 3 Sep 2020.
  4. ^ Simpson, Carlos (1998-10-09). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.
  5. ^ Brown, Ronald; Higgins, Philip J. (1981). "The equivalence of $\infty $-groupoids and crossed complexes". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 22 (4): 371–386.
  6. ^ Ara. "Sur les infinity-groupoïdes de Grothendieck et une variante infinity-catégorique" (PDF). Section 1.4.3. Archived (PDF) from the original on 19 Aug 2020.

Research articles

Applications in algebraic geometry

External links

This page was last edited on 9 September 2021, at 13:25
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