∞-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 fundemantal n-groupoid ${\displaystyle \Pi _{n}(Y)}$. Such a space can be found using the Whitehead tower.

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.

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

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

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 upto 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.

• On the homotopy hypothesis in dimension 3
• Note on the construction of globular weak omega-groupoids from types, topological spaces etc
• Higher Monodromy
• Higher Galois theory

• Homotopy types of algebraic varieties - Bertrand Toën