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Pardon if this question is vague or misguided, I'm a CS person who only dabbles in math.

In a groupoid all morphisms are isomorphisms. So then, any two objects with a morphism between them must be isomorphic.

So then isn't the only question that matters in a groupoid the question of whether two objects are connected at all? If they are connected they're isomorphic and in the same equivalence class, and if they're not then they're in different equivalence classes.

If you skeletalize a groupoid, it seems to me that you'd just get a bunch of groups sitting next to each other, not talking. I'm wondering -- is this correct, or is there a subtlety I'm missing? And if I am correct -- what's the practical benefit of talking about a groupoid, as opposed to just talking about a collection of groups?

James Gilles
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    For the benefit (see title) see for example this post. – Dietrich Burde May 24 '21 at 18:50
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    Thought I'd give a general comment: usually, the reason Mathematicians like generalising structures is because they show up in other fields and that's all we have to play with, so we would like a list of consequences of such structures so we can readily use them in other fields – Riemann'sPointyNose May 24 '21 at 19:23
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    Does this answer your question? How are groupoids richer structures than sets of groups? – Noah Schweber May 24 '21 at 20:03
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    I used groupoids under the name of "inverse semigroup" and "partial permutations". I was not studying the objects, but rather the morphisms. Permutations of five objects are interesting but the numbers 1,2,3,4,5 from the set {1,2,3,4,5} might all be equivalently dull. Similarly, the domains and ranges might be dull, but the way we permute them may be quite interesting. https://www.gap-system.org/Manuals/doc/ref/chap54_mj.html – Jack Schmidt May 24 '21 at 21:19
  • Thanks for the links! Didn't see those doing keyword searches beforehand. That answers my question, yes. – James Gilles May 24 '21 at 21:46
  • If you "skeletalize" a groupoid you just get a single group. In doing so you have discarded the information that the groupoid captured. One of the motivating examples for groupoids is the notion of the fundamental group of a topological space: that notion depends on the choice of a base point in the space. The notion of the fundamental groupoid allows us to abstract away from that choice. – Rob Arthan May 24 '21 at 21:48

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