# 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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### A comparison between the fundamental groupoid and the fundamental group

Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
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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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### 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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### Categorification of $\pi$?

Is there a categorification of $\pi$? I have to admit that this is a very vague question. Somehow it is motivated by this recent MO question, which made me stare at some digits and somehow forgot my ...
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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). ...
Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
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 ...