# Tagged Questions

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical ...

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### Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
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### Characterization of internal groupoids via pullbacks

The most intuitive way (for me) to define an internal groupoid is as an internal category with extra structure, namely an involution on the object of morphisms which "produces inverses". In Borceux ...
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### The Image of a Groupoid Morphism is Not a Subgroupoid

I am reading Brown's "Topology and Groupoids" book. In it he defines a subgroupoid as a subcategory of a groupoid that contains all the inverses for all the morphisms. A morphism of groupoids is a ...
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### Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
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This is similar to a question I asked recently, but this time specifically for groupoids. Suppose $f: A \rightarrow B$ is a groupoid morphism. Let $f^\ast: [B, \text{Set}] \rightarrow [A, \text{Set}]... 2answers 99 views ### Which constructions on a category are still interesting for a groupoid? By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ... 3answers 140 views ### From groups to groupoids. Let$\mathcal{G}$be a groupoid and$p$an object in$\mathcal{G}.$It is well known that the set${\rm Mor}_{\mathcal{G}}(p,p)$is a group. I would like to know if there is a way to recognize a ... 0answers 43 views ### Determination of a connected groupoid by its objects and by a set of automorphisms. [duplicate] One may readily show that a connected groupoid$G$is determined up to isomorphism by a group (one of the groups$\hom_G(x,x)$) and by a set (the set of all objects). This is the nature of the problem ... 1answer 54 views ### Groupoid-valued presheaf as a colimit of representables Is there a specific way to see a presheaf of groupoids as a colimit of representables ? As you can understand I'm looking for a similar result to the well-known fact that presheaves of sets are ... 0answers 73 views ### Torsors for 2-groups Let$\mathbb{G}$be a 2-group, by which I mean a strict monoidal category in which all objects are invertible (up to coherent isomorphisms) and all morphisms are invertible (strictly). What is the ... 2answers 220 views ### Which of these constructions are left adjoints? A groupoid can be regarded as a small category in which every arrow is an isomorphism A monoid can be regarded as a small category with only a single object A preorder can be regarded as a small ... 1answer 425 views ### Where to get help with Homotopy type theory? I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ... 1answer 95 views ### Should the first be the last by composition of paths? Given two paths$f,g:\mathbb{I}\rightarrow X$with$f\left(1\right)=g\left(0\right)$there is a composite$f.g$defined by$t\mapsto f\left(2t\right)$if$2t\leq1$and$t\mapsto g\left(2t-1\right)$... 3answers 492 views ### Statement about Homotopy in Brown's “Topology & Groupoids” I am trying to understand a statement in Brown's Topology and Groupoids, 7.2.5 (Corollary 1), page 270. Let's first have some preliminary remarks Let$X,Y$be topological spaces. The track groupoid ... 2answers 98 views ### Is a groupoid a universal algebra? I was trying to use the first isomorphism theorem on groupoids. From the wikipedia page I know that it holds for groups, rings and algebras. So are groupoids algebras? Or, in other words, does the ... 0answers 188 views ### Two definitions of equivariant sheaves Let$G$be a topological group. Here are two definitions of$G$-equivariant sheaves on a$G$-space$X$. (a) Define an$G$-equivariant sheaf by a sheaf$F$(étalé space) equipped with a$G$-action ... 2answers 470 views ### A comparison between the fundamental groupoid and the fundamental group Are there two path connected topological spaces$X,Y$such that the fundamental groupoid of$X$is not isomorphic to the fundamental groupoid of$Y$but the fundamental group of$X$is isomorphic to ... 1answer 407 views ### Existence of a certain functor$F:\mathrm{Grpd}\rightarrow\mathrm{Grp}$Let$\mathrm{Grpd}$denote the category of all groupoids. Let$\mathrm{Grp}$denote the category of all groups. Are there functors$F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, G\colon\mathrm{Grp}\...
It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to the action groupoid coming from a transitive action of some group \$...