# ∞-groupoid

Short description: Abstract homotopical model for topological spaces

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

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{align} \sigma_n:[n] \to [n+1]\\ \tau_n:[n] \to [n+1] \end{align} }

such that the globular relations hold

\displaystyle{ \begin{align} \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{align} }

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{align} s_n = X_\bullet(\sigma_n) \\ t_n = X_\bullet(\tau_n) \end{align} }

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 $\displaystyle{ \pi_{\leq n}(S^2) }$ 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 $\displaystyle{ \pi_{\gt n}(Y) = 0 }$ 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 $\displaystyle{ C_\bullet \in \text{Ch}_{\leq0}(\text{Ab}) }$.[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. Grothendieck. "Pursuing Stacks".
2. Maltsiniotis, Georges (2010), Grothendieck infinity groupoids and still another definition of infinity categories
3. Simpson, Carlos (1998-10-09). "Homotopy types of strict 3-groupoids". arXiv:math/9810059.
4. Brown, Ronald; Higgins, Philip J. (1981). "The equivalence of $\infty$-groupoids and crossed complexes" (in en). Cahiers de Topologie et Géométrie Différentielle Catégoriques 22 (4): 371–386.
5. 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.

•