# Tagged Questions

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### How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
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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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### Connected groupoids and action groupoids

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 ...
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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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### When is a morphism of $S$-groupoids a monomorphism?

According to "Champs algébriques" by Laumon and Moret-Bailley, and $S$-groupoid is a category $\mathscr{X}$ and a functor $a: \mathscr{X} \to (\mathrm{Aff}/S)$, where $(\mathrm{Aff}/S)$ is the ...
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### Relationship between functors

You will have to forgive me as I am very new to category theory - fifth of the way through Categories for a working mathematician. I'm interested in the following; Let $F:A \to B$ and $G:A \to C$ be ...