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      Questions tagged [groupoids]

      A groupoid is a category where all morphisms are invertible. This notion can also be seen as an extension of the notion of group. A motivating example is the fundamental groupoid of a topological space with respect to several base points, compared to the usual fundamental group.

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      What category of toposes is monadic over the 2-category of groupoids?

      Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the ...
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      191 views

      Automorphism group of a torsor

      Given a site $C$ and an object $U$, let $G$ be a sheaf of groups on this site and let $F$ be $G$-torsor, see the Stacks Project for the general definition. By restriction on the over category $C/U$ (...
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      230 views

      What is the category of algebras for the finitely supported measures monad?

      In this post, I was introduced to the monad of finitely supported measures. $HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad. I have three ...
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      1answer
      103 views

      Continuity of functions on étale groupoids

      Let $\mathcal G$ be an étale groupoid with a locally compact, Hausdorff unit space $\mathcal G^{(0)}$. If $U?\mathcal G$ is an open subset, which is Hausdorff in the induced topology, and if $f$ is ...
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      97 views

      Cencov's “categories of figures”

      In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...
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      1answer
      63 views

      Convolution product in an étale groupoid

      I am going through Sims - étale groupoids and their $C^*$ algebras and at Lemma 3.1.4. the author says that $f^**f\in C_c(G^{(0)})$ is supported on $s(supp(f))$ and $(f^**f)(s(\gamma))=|f(\gamma)|^2$ ...
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      197 views

      Fundamental groupoid and fibration

      In this post, it is said that a functor from the fundamental groupoid of a space $X$ (denoted by $\Pi(X)$) to the category $\mathrm{Vect}$ of vector spaces gives a flat vector bundle over $X$. But I ...
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      3k views

      Why did Voevodsky consider categories “posets in the next dimension”, and groupoids the correct generalisation of sets?

      Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...
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      113 views

      Does the 2 category of Groupoids Admit the Vector Space Monad?

      We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad. 3.2. Vector spaces. For a semiring S one can define the ...
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      1answer
      218 views

      Is Quillen's bracket a “universal enveloping” something?

      $\newcommand{\G}{\mathcal{G}}$ In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...
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      1answer
      209 views

      Slice theorem for proper groupoids

      Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$. Assume $G$ is étale, i.e., the source and range maps of $G$ are local ...
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      131 views

      Frobenius monads and groupoids

      For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ...
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      1answer
      426 views

      The étale topos of a scheme is the classifying topos of which groupoid?

      [Sent here from Math.StackExchange by suggestion of an user.] By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...
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      434 views

      Why study orbifolds? [closed]

      Question is as in the title. Why study orbifolds? I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. ...
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      1answer
      183 views

      Isotropy group of a Lie groupoid is a Lie group

      I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group. Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0$...

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