# Tagged Questions

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### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$ On ...
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### How to construct a free 2-group on a groupoid?

Let G be a groupoid. I'm wondering how to construct the free 2-group on G. By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$ equipped with a functor ...
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### Albrecht Fröhlich's text Groupoids, groupoid spaces and cohomology' (1965)

I am looking for Albrecht FrÃ¶hlich's unpublished text Groupoids, groupoid spaces and cohomology' (1965). In this text FrÃ¶hlich defines cohomology of a group with coefficients in a groupoid (this was ...
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### Groupoids vs. action groupoids

Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows. My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a ...
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### Equivalence and weak equivalence of groupoids

Let $A\rightrightarrows X$ be a groupoid, where $X$ is the set of objects and $A$ is the set of arrows. My favorite example of a groupoid is an action groupoid. If a group $G$ acts on the left on a ...
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### Connected groupoids and action groupoids

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to an action groupoid $G\ltimes X$ coming from a transitive action ...
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### Constructing a stack (gerbe) from a connected groupoid

Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid. Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$, and we have 5 maps: $s,t\colon A\to X$ (the source and the target, surjective), ...
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### Semidirect product of groupoids

I am trying to understand the semidirect product of groupoids, as defined in this answer by Theo Johnson-Freyd. Part of my difficulty is that although the definition of a 2-group makes sense, I am ...
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### What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where: $G$ is a Lie group, and $H$ ...
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### Groupoids and hypergroups

There are two generalizations of usual groups: groupoids, where the multiplication operation becomes "partial", and hypergroups, for which the result of multiplying two elements is a probability ...
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### Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...
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### Convolution algebras for double groupoids?

There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
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### Toposes (topoi) as classifying toposes of groupoids

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Buntz and Moerdijk have shown that if the topos has ...
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### A category being self-dual vs. it being a groupoid

What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference? To go from a self-duality $F$ on a ...
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### Characterizing Groupoids via Quotients?

A groupoid is a category in which all morphisms are invertible.(*) The groupoids form a very nice subclass of categories. The inclusion of the groupoids into the 2-category of small categories admits ...
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### Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...
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### How equivalent are the theories of reduced and groupal $\infty$-groupoids?

I hope that my question is sufficiently trivial that someone will be able to give me a pedantic answer, and not so trivial that no one takes the time to give an answer. My motivation for asking this ...
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### How to interpret topologically that the equalizer in Groupoids of ${\rm id}, {\rm id}: BG \rightrightarrows BG$ is $G/G$ (adjoint action)?

Let $G$ be a (discrete) group, and $1/G$ the corresponding groupoid with one object. Consider the diagram in (the 2-category) Groupoids: (This diagram is equivalent to the pair of parallel arrows ...
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### $\infty-$groupoid of $A_{\infty}$ algebras

Hello, Consider first the following $2-$groupoid of Algebras over $\mathbb{C}$. Objects are Algebras, $1-$morphisms are isomorphisms, and a $2-$morphism between the isoms $f$ and $g$ from $A$ to $B$ ...
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### What condition on a “bibundle between categories” generalizes “right-principal bibundle between groupoids”?

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...
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### Colimits of topological groupoids

Let $G$ and $H$ be two topological groupoids. Suppose that I have two morphisms $G \rightrightarrows H$ and I want to take the 2-coequalizer of these maps. I'd like an explicit description of (a ...
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### Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...
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### What is the 2-category whose 0-objects are Lie algebroids?

Recall the notion of Lie algebroid (n Lab, Wikipedia). One motivation for studying Lie algebroids is that they are infinitesimal versions of Lie groupoids, and Lie groupoids present stacks. In ...
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### Is a groupoid determined by its Hopfish algebra?

This is a follow up to my question What is the precise relationship between groupoid language and noncommutative algebra language?. I will briefly review some definitions; for details, a good place ...
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### What is the precise relationship between groupoid language and noncommutative algebra language?

I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose: objects are groupoids; 1-morphisms are (left-principal?) bibundles; 2-morphisms are bibundle ...
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### Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks. One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if ...
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### What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid?

I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) ...
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### How can I understand the “groupoid” quotient of a group action as some sort of “product”?

Recall the notion of groupoid (Wikipedia, nLab). An important construction of groupoids is as "action groupoids" for group actions. Namely, let $X$ be a groupoid and $G$ a group, and suppose that ...
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### Is there a “groupoid integral” with values in a groupoid?

Let $G = \{G_1 \rightrightarrows G_0\}$ be a finite groupoid, i.e. $G_1,G_0$ are both finite sets, and let $A$ be $\mathbb Q$-module. Regard $A$ as a discrete groupoid $A \rightrightarrows A$, and ...
Is it possible to split a given category $C$ up into its groupoid of isomorphisms and a category that resembles a poset? "Splitting up" should be that $C$ can be expressed as some kind of extension ...
This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf. Let $M$ be a ...