{{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).<ref>{{Cite web | url=https://ncatlab.org/nlab/show/Kan+complex#AsGrpds |title = Kan complex in nLab}}</ref> 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.<ref name=":0" />{{rp|2-3}}<ref>{{Citation|last=Maltsiniotis|first=Georges|date=2010|title=Grothendieck infinity groupoids and still another definition of infinity categories |arxiv=1009.2331 |citeseerx=10.1.1.397.2664}}</ref>
== Globular Groupoids == Alexander Grothendieck suggested in ''Pursuing Stacks''<ref name=":0">{{Cite web|last=Grothendieck|date=|title=Pursuing Stacks|url=https://thescrivener.github.io/PursuingStacks/|url-status=live|archive-url=https://web.archive.org/web/20200730015735/https://thescrivener.github.io/PursuingStacks/ps-online.pdf|archive-date=30 Jul 2020|access-date=2020-09-17|website=thescrivener.github.io}}</ref>{{rp|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 <math>\mathbb{G}</math>. This is defined as the category whose objects are finite ordinals <math>[n]</math> and morphisms are given by <math display="block">\begin{align} \sigma_n: [n] \to [n+1]\\ \tau_n: [n] \to [n+1] \end{align}</math> such that the globular relations hold <math display="block">\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}</math> These encode the fact that {{mvar|n}}-morphisms should not be able to ''see'' {{math|(''n'' + 1)}}-morphisms. When writing these down as a globular set <math>X_\bullet: \mathbb{G}^{op} \to \text{Sets}</math>, the source and target maps are then written as <math display="block">\begin{align} s_n = X_\bullet(\sigma_n) \\ t_n = X_\bullet(\tau_n) \end{align}</math> We can also consider globular objects in a category <math>\mathcal{C}</math> as functors <math display="block">X_\bullet\colon \mathbb{G}^{op} \to \mathcal{C} .</math> 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 <math>S^2</math> its associated homotopy <math>n</math>-type <math>\pi_{\leq n}(S^2)</math> can never be modeled as a strict globular groupoid for <math>n \geq 3</math>.<ref name=":0" />{{rp|445}}<ref>{{cite arXiv|last=Simpson|first=Carlos|author-link=Carlos Simpson|date=1998-10-09|title=Homotopy types of strict 3-groupoids|eprint=math/9810059}}</ref> This is because strict ∞-groupoids only model spaces with a trivial Whitehead product.<ref>{{Cite journal|last1=Brown|first1=Ronald|last2=Higgins|first2=Philip J.|date=1981|title=The equivalence of $\infty $-groupoids and crossed complexes|url=http://www.numdam.org/item/CTGDC_1981__22_4_371_0/|journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques|language=en|volume=22|issue=4|pages=371–386}}</ref>
== Examples ==
=== Fundamental ∞-groupoid === Given a topological space <math>X</math> there should be an associated '''fundamental ∞-groupoid''' <math>\Pi_{\infty} X</math> where the objects are points <math>x \in X</math>, {{nowrap|1-morphisms}} <math>f:x \to y</math> are represented as paths, {{nowrap|2-morphisms}} are homotopies of paths, {{nowrap|3-morphisms}} are homotopies of homotopies, and so on. From this ∞-groupoid we can find an <math>n</math>-groupoid called the '''fundamental <math>n</math>-groupoid''' <math>\Pi_n X</math> whose homotopy type is that of <math>\pi_{\leq n} X</math>.
Note that taking the fundamental ∞-groupoid of a space <math>Y</math> such that <math>\pi_{>n} Y = 0</math> is equivalent to the fundamental ''n''-groupoid <math>\Pi_n Y</math>. 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 <math>C_\bullet \in \text{Ch}_{\leq0}(\text{Ab})</math>.<ref>{{Cite thesis |last=Ara |first=Dimitri |date=2010 |title=Sur les ∞-groupoïdes de Grothendieck et une variante ∞-catégorique |type=PhD |publisher=Université Paris Diderot |url=http://www.normalesup.org/~ara/files/these.pdf|url-status=live|archive-url=https://web.archive.org/web/20200819191139/http://www.normalesup.org/~ara/files/these.pdf|archive-date=19 Aug 2020 |at=Section 1.4.3}}</ref> There is an associated globular groupoid. Intuitively, the objects are the elements in <math>C_0</math>, morphisms come from <math>C_0</math> through the chain complex map <math>d_1:C_1 \to C_0</math>, and higher <math>n</math>-morphisms can be found from the higher chain complex maps <math>d_n:C_n \to C_{n-1}</math>. We can form a globular set <math>\mathbb{C}_\bullet</math> with <math display="block">\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}</math> and the source morphism <math>s_n:\mathbb{C}_n \to \mathbb{C}_{n-1}</math> is the projection map <math display="block">pr:\bigoplus_{k=0}^{n}C_k \to \bigoplus_{k=0}^{n-1}C_k</math> and the target morphism <math>t_n: C_n \to C_{n-1}</math> is the addition of the chain complex map <math>d_n: C_n \to C_{n-1}</math> 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 <math>\Pi X = \Pi_{\leq 1} X</math> to the category of abelian groups, the category of <math>R</math>-modules, or some other abelian category. That is, a local system is equivalent to giving a functor <math display="block">\mathcal{L}: \Pi X \to \text{Ab}</math> 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 {{nowrap|∞-functor}} <math display="block">\mathcal{L}_\bullet: \Pi_\infty X \to D(\text{Ab})</math> with values in some derived category. This has the advantage of letting the higher homotopy groups <math>\pi_n X</math> to act on the higher local system, from a series of truncations. A toy example to study comes from the Eilenberg–MacLane spaces <math>K(A, n)</math>, 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 <math>\mathcal{L}_\bullet: \Pi_\infty X \to D(\text{Ab})</math> from their truncations <math>\Pi_n X</math> and the maps <math>\tau_{\leq n-1}: \Pi_n X \to \Pi_{n-1} X</math> whose fibers should be the categories of <math>n</math>-functors <math display="block">\Pi_n(K(\pi_n X, n)) \to D(\text{Ab})</math> Another advantage of this formalism is it allows for constructing higher forms of <math>\ell</math>-adic representations by using the etale homotopy type <math>\hat{\pi}(X)</math> of a scheme <math>X</math> and construct higher representations of this space, since they are given by functors <math display="block">\mathcal{L}:\hat{\pi(X)} \to D(\overline{\mathbb{Q}}_\ell)</math>
=== Higher gerbes === Another application of ∞-groupoids is giving constructions of ''n''-gerbes and ∞-gerbes. Over a space <math>X</math> an ''n''-gerbe should be an object <math>\mathcal{G} \to X</math> such that when restricted to a small enough subset <math>U \subset X</math>, <math>\mathcal{G}|_U \to U</math> 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 <math>\mathcal{G} \to X</math> such that over any open subset <math display="block">\mathcal{G}|_U \to U</math> 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 a site <math>\mathcal{X}</math>, e.g. <math display="block">p:\mathcal{C}\to \mathcal{X}</math> will give an example of a higher gerbe if the category <math>\mathcal{C}_U</math> lying over any point <math>U \in \operatorname{Ob}\mathcal{X}</math> is a non-empty category. In addition, it would be expected this category would satisfy some sort of descent condition.
== See also == {{Portal|Mathematics}} *Pursuing Stacks *''n''-group *Homotopy type theory *Core of a category *(∞, n)-category *Joyal's extension and lifting theorems
== References == {{reflist}}
=== Research articles === *{{cite journal |last1=Henry |first1=Simon |last2=Lanari |first2=Edoardo |title=On the homotopy hypothesis for 3-groupoids |journal=Theory and Applications of Categories |date=2023 |volume=39 |pages=735–768 |doi=10.70930/tac/897kfv5y|arxiv=1905.05625}} *{{cite journal |last1=Bourke |first1=John |title=Note on the construction of globular weak omega-groupoids from types, topological spaces etc |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2016 |volume=LVII |issue=4 |pages=281–294 |url=https://cahierstgdc.com/wp-content/uploads/2017/11/Bourke-57-4.pdf| arxiv=1602.07962}} *{{cite journal |last1=Polesello |first1=Pietro |last2=Waschkies |first2=Ingo |title=Higher monodromy |journal=Homology, Homotopy and Applications |date=2005 |volume=7 |issue=1 |pages=109–150 |doi=10.4310/HHA.2005.v7.n1.a7| arxiv=math/0407507}} *{{cite journal |last1=Hoyois |first1=Marc |title=Higher Galois theory |journal=Journal of Pure and Applied Algebra |date=1 July 2018 |volume=222 |issue=7 |pages=1859–1877 |doi=10.1016/j.jpaa.2017.08.010 |arxiv=1506.07155 }}
=== Applications in algebraic geometry ===
*{{cite web |citeseerx=10.1.1.607.9789 |title=Homotopy types of algebraic varieties |author-link=Bertrand Toën |first=Bertrand |last=Toën |url=https://www.aimath.org/WWN/motivesdessins/Toen.pdf}}
== Further reading == * https://mathoverflow.net/questions/404210/delooping-monoidal-infty-groupoids-into-infty-categories?rq=1
== External links == *{{nlab|id=infinity-groupoid}} *{{nlab|id=fundamental+infinity-groupoid|title=fundamental infinity-groupoid}} *{{cite arXiv|first=Georges|last=Maltsiniotis|title=Grothendieck ∞-groupoids, and still another definition of ∞-categories|eprint=1009.2331|year=2010 |class=math.CT|mode=cs2}} *{{citation|first=Marek|last=Zawadowski|title=Introduction to Test Categories|url=http://www-home.math.uwo.ca/~kkapulki/notes/test_categories.pdf|archive-url=https://web.archive.org/web/20150326182023/http://www-home.math.uwo.ca/~kkapulki/notes/test_categories.pdf|url-status=dead|archive-date=2015-03-26}} *{{citation |first=Tom |last=Lovering |date=2012 |title=Etale cohomology and Galois Representations |citeseerx=10.1.1.394.9850 }}
{{Category theory}} {{Foundations-footer}}
{{DEFAULTSORT:Infinity groupoid}} Category:Foundations of mathematics Category:Higher category theory Category:Homotopy theory Category:Simplicial sets