The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat^1-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of a foliated bundle.
We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps.
We prove that any homotopy type can be recovered canonically from its associated weak omega-groupoid. This implies that the homotopy category of CW-complexes can be embedded in the homotopy category of Batanin's weak higher groupoids.
We show that the category of presheaves of simplicial groupoids on a site $\mathcal{C}$ is a right proper simplicial model category. We define $G$-torsor of presheaf of 2-groupoids $G$, presheaf of simplicial groups $G$ and presheaf of simplicial groupoids $G$ on a site $\mathcal{C}$ and classify $G$-torsors by the homotopy classes $[\ast, \bar{W}G]$.
This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural `filling condition'. Any such double groupoid characteristically has associated to it `homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of `weak equivalence' between such double groupoids, and a corresponding `homotopy category…
It is shown that the category of presheaves of simplicial groupoids and the category of presheaves of 2-groupoids have Quillen closed model structures. Furthermore, their homotopy categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively.
The purpose of this text is the study of the class of homotopy types which are modelized by strict \infty-groupoids. We show that the homotopy category of simply connected \infty-groupoids is equivalent to the derived category in homological degree greater or equal to 2 of abelian groups. We deduce that the simply connected homotopy types modelized by strict \infty-groupoids are precisely the products of Eilenberg-Mac Lane spaces. We also briefly study 3-categories with weak inverses. We finish by two questions about the problem suggested by the title of this text.
We present a slight variation on a notion of weak \infty-groupoid introduced by Grothendieck in Pursuing Stacks and we study the homotopy theory of these \infty-groupoids. We prove that the obvious definition for homotopy groups of Grothendieck \infty-groupoids does not depend on any choice. This allows us to give equivalent characterizations of weak equivalences of Grothendieck \infty-groupoids, generalizing a well-known result for strict \infty-groupoids. On the other hand, given a model category M in which every object is fibrant, we construct, following Grothendieck, a fundamental \infty-g…
We use the n-globe with its skeletal filtration to define the fundamental globular omega--groupoid of a filtered space; the proofs use an analogous fundamental cubical omega--groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular omega-groupoid on one element of dimension n is the fundamental crossed complex of the n-globe.
We outline the main features of the definitions and applications of crossed complexes and cubical $ω$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.
This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition…
We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the étale topos of a scheme, to its refinement …
We look at strict $n$-groupoids and show that if $\Re$ is any realization functor from the category of strict $n$-groupoids to the category of spaces satisfying a minimal property of compatibility with homotopy groups, then there is no strict $n$-groupoid $G$ such that $\Re (G)$ is the $n$-type of $S^2$ (for $n\geq 3$). At the end we speculate on how one might fix this problem by introducing a notion of ``snucategory'', a strictly associative $n$-category with only weak units.
We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category of weak infinity groupoids, whose objects are all fibrant and which is in a precise sense "freely generated by an object". We show that all those semi model categories are Quillen equivalent together and Quillen to the model category of spaces. A general procedure is given to produce such coherator, and several explicit examples are presented: one…
A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to presentability of proper regular Lie groupoids.
We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.
This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace higher homotopy groups in quasi-category homotopy theory.
This belongs to a series of papers devoted to the study of the cohomology of classifying spaces of Lie groupoids. Our aim here is to introduce and study the notion of representation up to homotopy of Lie groupoids, the resulting derived category, and to show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bott's spectral sequence converging to the cohomology of classifying spaces from Lie groups to Lie groupoids.
Omar Antol{\'ı}n Camarena pointed out a gap in the proofs in \cite{Brown15,BrownJordan} of a condition for the Phragmen--Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.
The paper contains an application of van Kampen theorem for groupoids for computation of homotopy types of certain class of non-compact foliated surfaces obtained by gluing at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.
To a "stable homotopy theory" (a presentable, symmetric monoidal stable $\infty$-category), we naturally associate a category of finite étale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic $\mathbf{E}_\infty$-algebra of topological modular forms is trivial and that the Galois group of $K(n)$-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the …
This very speculative sketch suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds.
Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-group…
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or "invertible" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard mod…
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
Given an infinity-category C, one can naturally construct an infinity-category Fam(C) of families of objects in C indexed by infinity-groupoids. An ordinary categorical version of this construction was used by Borceux and Janelidze in the study of generalized covering maps in categorical Galois theory. In this paper, we develop the homotopy theory of such "parametrized families" as generalization of the classical homotopy theory of spaces. In particular, we study homotopy-theoretical constructions that arise from the fundamental infinity-groupoids of families in an infinity-category. In the sa…
Homotopy 3-types can be modelled algebraically by Tamsamani's weak 3-groupoids as well as, in the path-connected case, by cat^2-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using the Tamsamani's model. Both are then related to Gray groupoids.
We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups.
We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.
Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.
The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.
An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analog…
By homotopy linear algebra we mean the study of linear functors between slices of the $\infty$-category of $\infty$-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into $\infty$-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez-Hoffnung-Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over $\infty$-groupoids; we hope that they can also be of i…
This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the étale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects given by homotopy fixed point constructions, but only up to pro-equivalence. The homotopy fixed point spaces define finite Galois descent for simplicial presheaves (and their relatives) over a field, but a pro-categorical construction is a necessary second step for passage from finite descent conditions to full homotopy descent in a Galois coho…
Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showi…
The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to n-cubes of pointed spaces.
We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic…
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicial ly enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions; (ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the o…
For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serre's finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and…
We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction of an almost free action of a Lie group G on a smooth …
In this paper we construct a symmetric monoidal closed model category of coherently commutative Picard groupoids. We construct another model category structure on the category of (small) permutative categories whose fibrant objects are (permutative) Picard groupoids. The main result is that the Segal's nerve functor induces a Quillen equivalence between the two aforementioned model categories. Our main result implies the classical result that Picard groupoids model stable homotopy one-types.
The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.
It is well-known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological groupoids to groupoids. This article shows that for a map $Y \to X$ with certain relative local connectivity assumptions, the fibrewise fundamental groupoid can also be lifted to a topological groupoid over the space $X$. This allows the construction of a simply-connected covering space in the setting of fibre…
In previous work, the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if $L/k$ is a finite Galois extension of fields with Galois group $G$, there is a functor $c_{L/k}^*$ from the $G$-equivariant stable homotopy category to the stable motivic homotopy category over $k$ such that $c_{L/k}^*(G/H_+) = Spec(L^H)_+$. We proved that when $k$ is a real closed field and $L=k[i]$, the restriction of $c_{L/k}^*$ to the $η$-complete subcategory is full and faithful. Here we "uncomplete" this theorem so that it applies to $c_{L/k}^*$ itsel…
We calculate the higher homotopy groups of the Deligne-Getzler infinity-groupoid associated to a nilpotent L-infinity algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.
Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined …
By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.
Informally, an orbifold is a smooth space whose points may have finitely many internal symmetries. Formally, however, the notion of orbifold has been presented in a number of different guises -- from Satake's V-manifolds to Moerdijk and Pronk's proper étale groupoids -- which do not on their face resemble the informal definition. The reason for this divergence between formalism and intuition is that the points of spaces cannot have internal symmetries in traditional, set-level foundations. The extra data of these symmetries must be carried around and accounted for throughout the theory. More d…
We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG to classifiying spaces of certain higher groupoids. In particular, to every fusion category C we attach the 3-groupoid BrPic(C) of invertible C-bimodule categories, called the Brauer-Picard groupoid of C, such that equivalence classes of G-extensions of C are in bijection with homotopy classes of maps from BG to the classifying space of BrPic(C). This gives…
The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space $X$. The homotopy type of $\mathfrak{X}$ should be the classifying space $B \mathbb{X}$. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this note is to give an elementary construction of such a homotopy-type functor.
We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class groupoid $\mathrm{MCG}_{M}^{A}$, and from the full subgroupoid of the motion groupoid $\mathrm{Mot}_{M}^{A}$, whose objects are homotopically 1-finitely generated. We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to \mathbf{Vect}$, one for each finite gro…
