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 equivalent to spaces up to homotopy.[2]: 2–3 [3]

Globular Groupoids

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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 n-morphisms should not be able to see (n + 1)-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

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Fundamental ∞-groupoid

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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 ∞-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

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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

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Higher local systems

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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

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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 a 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

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References

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  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 (2010), Grothendieck infinity groupoids and still another definition of infinity categories, arXiv:1009.2331, CiteSeerX 10.1.1.397.2664
  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, Dimitri (2010). Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique (PDF) (PhD). Université Paris Diderot. Section 1.4.3. Archived (PDF) from the original on 19 Aug 2020.

Research articles

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Applications in algebraic geometry

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