∞-groupoid explained

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.

Globular Groupoids

Alexander Grothendieck suggested in Pursuing Stacks^[2] 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

G . This is defined as the category whose objects are finite ordinals [n] and morphisms are given by$$\backslash begin\backslash sigma\_n:\; [n]\; \backslash to\; [n+1]\backslash \backslash \backslash tau\_n:\; [n]\; \backslash to\; [n+1]\backslash end$$such that the globular relations holdThese encode the fact that n-morphisms should not be able to see (n + 1)-morphisms. When writing these down as a globular set X _{\bullet:} G ^{op} \to Sets , the source and target maps are then written as$$\backslash begins\_n\; =\; X\_\backslash bullet(\backslash sigma\_n)\; \backslash \backslash t\_n\; =\; X\_\backslash bullet(\backslash tau\_n)\backslash end$$We can also consider globular objects in a category l{C} as functors$$X\_\backslash bullet\backslash colon\; \backslash mathbb^\; \backslash to\; \backslash mathcal\; .$$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 S ^{2} its associated homotopy n -type \pi _{\leq} (S ^{2)} can never be modeled as a strict globular groupoid for n \geq 3 .^[3] This is because strict ∞-groupoids only model spaces with a trivial Whitehead product.^[4]

Examples

Fundamental ∞-groupoid

Given a topological space

X there should be an associated fundamental ∞-groupoid \Pi _{infty} X where the objects are points x \in X , f:x \to y are represented as paths, are homotopies of paths, are homotopies of homotopies, and so on. From this ∞-groupoid we can find an n -groupoid called the fundamental n -groupoid \Pi _{n} X whose homotopy type is that of \pi _{\leq} X .

Note that taking the fundamental ∞-groupoid of a space

Y such that \pi _{>n} Y = 0 is equivalent to the fundamental n-groupoid \Pi _{n} Y . 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

C _{\bullet} \in Ch _{\leq0} ( Ab ) .^[5] There is an associated globular groupoid. Intuitively, the objects are the elements in C _{0} , morphisms come from C _{0} through the chain complex map d _{1:C} _{1} \to C _{0} , and higher n -morphisms can be found from the higher chain complex maps d _{n:C} _{n} \to C _{n-1} . We can form a globular set C _{\bullet} withand the source morphism s _{n:C} _{n} \to C _{n-1} is the projection map$$pr:\backslash bigoplus\_^C\_k\; \backslash to\; \backslash bigoplus\_^C\_k$$and the target morphism t _{n:} C _{n} \to C _{n-1} is the addition of the chain complex map d _{n:} C _{n} \to C _{n-1} 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

\Pi X = \Pi _{\leq} X to the category of abelian groups, the category of R -modules, or some other abelian category. That is, a local system is equivalent to giving a functor$$\backslash mathcal:\; \backslash Pi\; X\; \backslash to\; \backslash text$$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 $$\backslash mathcal\_\backslash bullet:\; \backslash Pi\_\backslash infty\; X\; \backslash to\; D(\backslash text)$$with values in some derived category. This has the advantage of letting the higher homotopy groups \pi _{n} X to act on the higher local system, from a series of truncations. A toy example to study comes from the Eilenberg–MacLane spaces K(A, n) , 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 l{L} _{\bullet:} \Pi _{infty} X \to D( Ab ) from their truncations \Pi _{n} X and the maps \tau _{\leq} : \Pi _{n} X \to \Pi _{n-1} X whose fibers should be the categories of n -functors$$\backslash Pi\_n(K(\backslash pi\_n\; X,\; n))\; \backslash to\; D(\backslash text)$$Another advantage of this formalism is it allows for constructing higher forms of \ell -adic representations by using the etale homotopy type \hat{\pi}(X) of a scheme X and construct higher representations of this space, since they are given by functors$$\backslash mathcal:\backslash hat\; \backslash to\; D(\backslash overline\_\backslash ell)$$

Higher gerbes

Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes. Over a space

X an n-gerbe should be an object l{G} \to X such that when restricted to a small enough subset U \subset X , l{G}| _{U} \to U 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 l{G} \to X such that over any open subset$$\backslash mathcal|\_U\; \backslash to\; U$$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 l{X} , e.g.$$p:\backslash mathcal\backslash to\; \backslash mathcal$$will give an example of a higher gerbe if the category l{C} _{U} lying over any point U \in \operatorname{Ob} l{X} is a non-empty category. In addition, it would be expected this category would satisfy some sort of descent condition.

See also

• Pursuing Stacks
• n-group
• Groupoid
• Homotopy type theory

References

Research articles

• 1905.05625 . On the homotopy hypothesis in dimension 3. Henry . Simon . Lanari . Edoardo . 2019 . math.CT .
• 1602.07962. Note on the construction of globular weak omega-groupoids from types, topological spaces etc. Bourke . John . 2016 . math.CT .
• math/0407507. Higher Monodromy. Polesello . Pietro . Waschkies . Ingo . 2004 .
• 1506.07155. Higher Galois theory. Hoyois . Marc . 2015 . math.CT .

Applications in algebraic geometry

• Web site: 10.1.1.607.9789 . Homotopy types of algebraic varieties . Bertrand Toën . Bertrand . Toën .

External links

• • Georges. Maltsiniotis. Grothendieck ∞-groupoids, and still another definition of ∞-categories. 1009.2331. 2010 . math.CT. cs2.

Notes and References

1. Web site: Kan complex in nLab.
2. Web site: Grothendieck. Pursuing Stacks. live. https://web.archive.org/web/20200730015735/https://thescrivener.github.io/PursuingStacks/ps-online.pdf. 30 Jul 2020. 2020-09-17. thescrivener.github.io.
3. Simpson. Carlos. 1998-10-09. Homotopy types of strict 3-groupoids. math/9810059.
4. Brown. Ronald. Higgins. Philip J.. 1981. The equivalence of $\infty $-groupoids and crossed complexes. Cahiers de Topologie et Géométrie Différentielle Catégoriques. en. 22. 4. 371–386.
5. Ara . Dimitri . 2010 . Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique . PhD . Université Paris Diderot . live. https://web.archive.org/web/20200819191139/http://www.normalesup.org/~ara/files/these.pdf. 19 Aug 2020 . Section 1.4.3.