Articles of groupoids

covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between covering spaces of $X$ and actions of the fundamental groupoid $\Pi_1(X)$ on sets. I can see a functor from covering spaces to […]

Conjugacy Class of Isomorphisms Between Two Isomorphic Groups Definition

In Spanier’s algebraic topology book, in section 1.7 about the fundamental groupoid, he claims that if $A$ and $B$ are objects in the same component of a groupoid $\mathcal{G}$ (meaning that $hom_{\mathcal{G}}(A,B)\neq\varnothing)$, then the set $$\{F(f)\in hom_{Grp}(hom_{\mathcal{G}}(A,A),hom_{\mathcal{G}}(B,B))|f\in hom_{\mathcal{G}}(A, B)\}$$ is a “conjugacy class of isomorphisms in $hom_{Grp}(hom_{\mathcal{G}}(A,A),hom_{\mathcal{G}}(B,B))$”, where $F$ is the functor from the groupoid […]

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 animosity about this branch of mathematics, wondering if there is a connection to the branches I love so […]

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May’s A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let $\mathcal{O} = \{ U \}$ be a cover of a space $X$ by path connected open subsets such that the intersection of finitely many subsets in $\mathcal{O}$ is again […]

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 the fundamental group of $Y$ ? I guess that there exists such a pair of topological spaces. I don’t know an example […]

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 groupoid counts as $1/n!$ if it has $n!$ symmetries) equals $e$. He then says that this leads to a […]

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, it preserves only the structure of being a set. When $X$ is finite, what structure can the alternating group be said […]