definition of a groupoid - Mathematics Stack Exchange
Category-Theoretic Definition: A groupoid $\underline {G}$ is a small category (i.e. its class of objects $\underline {G}$ and its class of morphisms $\mathrm {Hom}_ {\underline {G}}$ are sets) in which …
Confused about the definition of a group as a groupoid with one object.
A groupoid is defined to be a category where every morphism is an isomorphism. So sometimes a group is said to just be a groupoid with one object. When I try to make sense of this, I denote the s...
differential geometry - Can the isotropy groups of a Lie groupoid be ...
Apr 22, 2025 · Our groupoid topologically integrates the Lie Algebroid $\ker (\alpha)$, but may not smoothly integrate the Lie Algebroid as $\mathcal {G}$ may not admit a smooth structure, so it may …
What is the canonical action of a groupoid on its unit space?
May 23, 2024 · But I have only seen the definition of a transformation groupoid for a group action. So what does the transformation groupoid of a groupoid action look like? The set of all bisections of a …
What is an example of a groupoid which is not a semigroup?
May 20, 2018 · I know that groupoid refers to an algebraic structure with a binary operation. The only necessary condition is closure. However, I couldn't find any easy-to-understand example of a …
category theory - Is there a nice characterization of which categories ...
May 29, 2025 · Can we give conditions on a category which characterize when it is equivalent to a presheaf category on a groupoid? Note that the nLab quotes some results characterizing presheaf …
category theory - Connected groupoids and action groupoids ...
A connected groupoid A can be written as an action groupoid for many different groups G. All the groupoid determines is H, the group of automorphisms of any object in the groupoid, and the index of …
quotient groupoid as the colimit in the infinite category Grpd
Jul 18, 2024 · For a group G G acting on a set X X, there is set theoretic quotient X/G X / G defined as the colimit of the following diagram in the category Set S e t: But this set forgets how two elements in …
What is a pointed connected groupoid? - Mathematics Stack Exchange
Aug 27, 2022 · A pointed groupoid is a groupoid with a chosen point (= object). The category of groupoids has a terminal object and you can see this as a special case of the general notion of …
Recovering a topological space from its fundamental groupoid
Jan 10, 2024 · Given only the fundamental groupoid of a topological space X, we can recover the underlying set of X since objects of the groupoid (as a category) are precisely the elements of X. I …