# 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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### What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words,...
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### Purely combinatorial proof that$(e^x)' = e^x$

At the beginning of Week 300 of John Baez's blog, Baez gives a proof that the "number" of finite sets (more specifically, the cardinality of the groupoid of all finite sets, where an object in the ...
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### 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 ...
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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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### 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 ...
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### How are groupoids richer structures than sets of groups?

This has been bugging me for quite some time: My intuition with categories is, that I can simply identify isomorphic objects. It does for example not matter, whether the entries in a sudoku are the ...
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### definition of a groupoid

Notation: Underlining $\underline{G}$ denotes a category and $\underline{G}(x,y)$ the class of morphisms from $x$ to $y$. On the Wiki page about groupoids, it is written (I write here my own more ...
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### 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? ...
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### The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two path-...
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### 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 ...
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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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### Conjugation in a groupoid

In a group $G$ , the conjugation of any element $g \in G$ by $h$ defines an inner automorphism $\psi_h : g \to hgh^{-1}$. If one considers instead a connected groupoid $\mathcal{G}$, is there a ...
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### 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 ...
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### 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 ...
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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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### Fundamental groupoid of a contractible space

I read that the fundamental groupoid of a contractible space is indiscrete. How can one show this? I found this as an exercise here.
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### Constructing the classifying space for a groupoid G

I am working with the classifying space BG of the groupoid G. One definition is as follows: $$BG = \bigsqcup _n (G_n \times \Delta ^n) /(d_i(g),x) \sim (g, \delta _i (x)).$$ Where the $d_i$ are ...
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### Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
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### can the statement “a simplicial set is the nerve of a category if and only if it satisfies a horn-filling condition” be tweaked for groupoids?

For some reason I convinced myself that a simplicial set (or maybe I mean directly Kan complex) is homotopy equivalent to the nerve of a groupoid if and only if it has no higher homotopy groups. Is ...
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### Topological Space as an $(\infty,0)$-category

Given a topological space $X$, we may wish to consider it as an $(\infty,0)$-category, where the objects are the points of the space, the 1-morphisms are continuous paths between points, the 2-...
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### If functions compose both ways to make automorphisms, are they isomorphisms?

Let's say that we have morphisms $f:A \to B$ and $g : B \to A$ such that $f \circ g$ and $g \circ f$ are both automorphisms (an automorphism is a morphism that is both iso and endo). Are $f$ and $g$ ...
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### Group of Isomorphisms of a Groupoid

Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category. First can someone tell me ...
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### Pushforward of a representation?

Suppose that $G$ is a finite group, and $G/N$ is a quotient. Given a representation of $G$, is there a "natural" way to construct a representation on $G/N$? (I.e. a pushforward representation, ...
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### Why should the source and the target map of a Lie groupoid be submersions?

In the definition of a Lie groupoid, the source and the target maps are required to be submersions. I want to know the reason for that. I write down definitions below. See also https://en.wikipedia....
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### In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
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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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### Etale groupoid and Morita equivalence

Let $\mathcal{G}=(G_{0},G_{1})$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $\mathcal{G}$ is called etale if both the source and target maps  s,t:G_{1}...