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.
[2]: 2–3 [3] 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
There was hope originally that such a strict model would be sufficient for homotopy theory, but there is evidence suggesting otherwise.
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.
Note that taking the fundamental ∞-groupoid of a space
Such a space can be found using the Whitehead tower.
-morphisms can be found from the higher chain complex maps
is the addition of the chain complex map
This forms a globular groupoid giving a wide class of examples of strict globular groupoids.
One of the basic theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid
That is, a local system is equivalent to giving a functor
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
Another advantage of this formalism is it allows for constructing higher forms of
-adic representations by using the etale homotopy type
and construct higher representations of this space, since they are given by functors
Another application of ∞-groupoids is giving constructions of n-gerbes and ∞-gerbes.
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
Because the nerve of a category can be used to construct an arbitrary homotopy type, a functor over a site
will give an example of a higher gerbe if the category
In addition, it would be expected this category would satisfy some sort of descent condition.