∞-groupoid
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category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses
Kan complex In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are ...
es which are fibrant objects in the category of
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
s (with the standard
model structure In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
). It is an
∞-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
generalization of a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
, a category in which every morphism is an isomorphism. The
homotopy hypothesis In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give mod ...
states that ∞-groupoids are spaces.


Globular Groupoids

Alexander Grothendieck suggested in ''
Pursuing Stacks ''Pursuing Stacks'' (french: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the w ...
'' that there should be an extraordinarily simple model of ∞-groupoids using
globular set In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets X_0, X_1, X_2, \dots equipped with pairs of functions s_n, t_n: X_n \to X_ such that * s_n \c ...
s, originally called hemispherical complexes. These sets are constructed as presheaves on the globular category \mathbb. This is defined as the category whose objects are finite ordinals /math> and morphisms are given by
\begin \sigma_n: \to +1\ \tau_n: \to +1\end
such that the globular relations hold
\begin \sigma_\circ\sigma_n &= \tau_\circ\sigma_n \\ \sigma_\circ\tau_n &= \tau_\circ\tau_n \end
These 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:\mathbb^ \to \text, the source and target maps are then written as
\begin s_n = X_\bullet(\sigma_n) \\ t_n = X_\bullet(\tau_n) \end
We can also consider globular objects in a category \mathcal as functors
X_\bullet\colon \mathbb^ \to \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_(S^n) can never be modeled as a strict globular groupoid for n \geq 3. This is because strict ∞-groupoids only model spaces with a trivial
Whitehead product In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in . The relevant MSC code is: 55Q15, Whitehead products and generalizations. Definition ...
.


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 1-morphisms 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 n-groupoid called the fundamental n-groupoid \Pi_n(X) whose homotopy type is that of \pi_(X). Note that taking the fundamental ∞-groupoid of a space Y such that \pi_(Y) = 0 is equivalent to the fundamental n-groupoid \Pi_n(Y). Such a space can be found using the
Whitehead tower In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
.


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 \text_(\text). 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_. We can form a globular set \mathbb_\bullet with
\begin \mathbb_0 =& C_0 \\ \mathbb_1 =& C_0\oplus C_1 \\ &\cdots \\ \mathbb_n =& \bigoplus_^n C_k \end
and the source morphism s_n:\mathbb_n \to \mathbb_ is the projection map
pr:\bigoplus_^C_k \to \bigoplus_^C_k
and the target morphism t_n: C_n \to C_ is the addition of the chain complex map d_n:C_n \to C_ 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 system In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology ...
s is that they can be equivalently described as a functor from the
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a to ...
\Pi(X) = \Pi_(X) to the category of Abelian groups, the category of R-modules, or some other
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
. That is, a local system is equivalent to giving a functor
\mathcal:\Pi(X) \to \text
generalizing such a definition requires us to consider not only an abelian category, but also its
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
. A higher local system is then an ∞-functor
\mathcal_\bullet:\Pi_\infty(X) \to D(\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 space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
s K(A, n), or by looking at the terms from the
Whitehead tower In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
of a space. Ideally, there should be some way to recover the categories of functors \mathcal_\bullet:\Pi_\infty(X) \to D(\text) from their truncations \Pi_n(X) and the maps \tau_:\Pi_n(X) \to \Pi_(X) whose fibers should be the categories of n-functors
\Pi_n(K(\pi_n(X),n)) \to D(Ab)
Another advantage of this formalism is it allows for constructing higher forms of \ell-adic representations by using the etale homotopy type \hat(X) of a scheme X and construct higher representations of this space, since they are given by functors
\mathcal:\hat \to D(\overline_\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 \mathcal \to X such that when restricted to a small enough subset U \subset X, \mathcal, _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 \mathcal \to X such that over any open subset
\mathcal, _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 \mathcal, e.g.
p:\mathcal\to \mathcal
will give an example of a higher gerbe if the category \mathcal_U lying over any point U \in \text(\mathcal) is a non-empty category. In addition, it would be expected this category would satisfy some sort of descent condition.


See also

*
Pursuing Stacks ''Pursuing Stacks'' (french: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the w ...
*
N-group (category theory) In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander G ...
*
Groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial functi ...
*
Homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...


References


Research articles

* 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


Applications in algebraic geometry


Homotopy types of algebraic varieties
- Bertrand Toën


External links

* * *
Etale cohomology and Galois Representations
{{DEFAULTSORT:Infinity groupoid Foundations of mathematics Higher category theory Homotopy theory Simplicial sets