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# ∞-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.[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 ${\displaystyle \mathbb {G} }$. 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. When writing these down as a globular set ${\displaystyle X_{\bullet }:\mathbb {G} ^{op}\to {\text{Sets}}}$, the source and target maps are then written as

{\displaystyle {\begin{aligned}s_{n}=X_{\bullet }(\sigma _{n})\\t_{n}=X_{\bullet }(\tau _{n})\end{aligned}}}

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. It turns out for ${\displaystyle S^{2}}$ its associated homotopy ${\displaystyle n}$-type ${\displaystyle \pi _{\leq n}(S^{n})}$ can never be modeled as a strict globular groupoid for ${\displaystyle n\geq 3}$.[2]: 445 [4] This is because strict ∞-groupoids only model spaces with a trivial Whitehead product.[5]

## Examples

### Fundamental ∞-groupoid

Given a topological space ${\displaystyle X}$ there should be an associated fundamental ∞-groupoid ${\displaystyle \Pi _{\infty }(X)}$ where the objects are points ${\displaystyle x\in X}$ 1-morphisms ${\displaystyle f:x\to y}$ 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 ${\displaystyle n}$-groupoid called the fundamental ${\displaystyle n}$-groupoid ${\displaystyle \Pi _{n}(X)}$ whose homotopy type is that of ${\displaystyle \pi _{\leq n}(X)}$.

Note that taking the fundamental ∞-groupoid of a space ${\displaystyle Y}$ such that ${\displaystyle \pi _{>n}(Y)=0}$ is equivalent to the fundamental n-groupoid ${\displaystyle \Pi _{n}(Y)}$. 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 ${\displaystyle C_{\bullet }\in {\text{Ch}}_{\leq 0}({\text{Ab}})}$.[6] There is an associated globular groupoid. Intuitively, the objects are the elements in ${\displaystyle C_{0}}$, morphisms come from ${\displaystyle C_{0}}$ through the chain complex map ${\displaystyle d_{1}:C_{1}\to C_{0}}$, and higher ${\displaystyle n}$-morphisms can be found from the higher chain complex maps ${\displaystyle d_{n}:C_{n}\to C_{n-1}}$. We can form a globular set ${\displaystyle \mathbb {C} _{\bullet }}$ with

${\displaystyle {\begin{matrix}\mathbb {C} _{0}=&C_{0}\\\mathbb {C} _{1}=&C_{0}\oplus C_{1}\\&\cdots \\\mathbb {C} _{n}=&\bigoplus _{k=0}^{n}C_{k}\end{matrix}}}$

and the source morphism ${\displaystyle s_{n}:\mathbb {C} _{n}\to \mathbb {C} _{n-1}}$ is the projection map

${\displaystyle pr:\bigoplus _{k=0}^{n}C_{k}\to \bigoplus _{k=0}^{n-1}C_{k}}$

and the target morphism ${\displaystyle t_{n}:C_{n}\to C_{n-1}}$ is the addition of the chain complex map ${\displaystyle d_{n}:C_{n}\to C_{n-1}}$ 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

One of the basic theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid ${\displaystyle \Pi (X)=\Pi _{\leq 1}(X)}$ to the category of Abelian groups, the category of ${\displaystyle R}$-modules, or some other abelian category. That is, a local system is equivalent to giving a functor

${\displaystyle {\mathcal {L}}:\Pi (X)\to {\text{Ab}}}$

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

${\displaystyle {\mathcal {L}}_{\bullet }:\Pi _{\infty }(X)\to D({\text{Ab}})}$

with values in some derived category. This has the advantage of letting the higher homotopy groups ${\displaystyle \pi _{n}(X)}$ to act on the higher local system, from a series of truncations. A toy example to study comes from the Eilenberg–MacLane spaces ${\displaystyle K(A,n)}$, 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 ${\displaystyle {\mathcal {L}}_{\bullet }:\Pi _{\infty }(X)\to D({\text{Ab}})}$ from their truncations ${\displaystyle \Pi _{n}(X)}$ and the maps ${\displaystyle \tau _{\leq n-1}:\Pi _{n}(X)\to \Pi _{n-1}(X)}$ whose fibers should be the categories of ${\displaystyle n}$-functors

${\displaystyle \Pi _{n}(K(\pi _{n}(X),n))\to D(Ab)}$

Another advantage of this formalism is it allows for constructing higher forms of ${\displaystyle \ell }$-adic representations by using the etale homotopy type ${\displaystyle {\hat {\pi }}(X)}$ of a scheme ${\displaystyle X}$ and construct higher representations of this space, since they are given by functors

${\displaystyle {\mathcal {L}}:{\hat {\pi (X)}}\to D({\overline {\mathbb {Q} }}_{\ell })}$

### Higher gerbes

Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes. Over a space ${\displaystyle X}$ an n-gerbe should be an object ${\displaystyle {\mathcal {G}}\to X}$ such that when restricted to a small enough subset ${\displaystyle U\subset X}$, ${\displaystyle {\mathcal {G}}|_{U}\to U}$ 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 ${\displaystyle {\mathcal {G}}\to X}$ such that over any open subset

${\displaystyle {\mathcal {G}}|_{U}\to U}$

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 an site ${\displaystyle {\mathcal {X}}}$, e.g.

${\displaystyle p:{\mathcal {C}}\to {\mathcal {X}}}$

will give an example of a higher gerbe if the category ${\displaystyle {\mathcal {C}}_{U}}$ lying over any point ${\displaystyle U\in {\text{Ob}}({\mathcal {X}})}$ is a non-empty category. In addition, it would be expected this category would satisfy some sort of descent condition.

## 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.

### Applications in algebraic geometry

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