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Stream: theory: category theory

Topic: Fibered groups and their actions on categories

Brendan Murphy (Nov 01 2023 at 01:42):

By a "fibered group" I mean a function $p : G \to J$ where each fiber $f^{-1}(j)$ has been equipped with the structure of a group. For fixed base $J$ this is a group object in $\mathsf{Set}/J$ (and we can also understand it as a functor from the discrete category $J$ to $\mathsf{Grp}$). There's an obvious notion of an action of a fibered group on a family $f : X \to J$ (fiberwise). For fibered groups $p : G \to J, q : H \to J'$ and families $f : X \to J, g : Y \to J'$ with a left action of $p$ on $f$ and a right action of $q$ on $g$ we can make sense of a $(p, q)$-biaction on a span $X \leftarrow Z \rightarrow Y$; one way of defining this is to think of the right action as a left action of the fiberwise opposite group $q^{\mathrm{op}} : H^{\mathrm{op}} \to J'$, then form the external product of fibered groups $p \times q^{\mathrm{op}} : G \times H^{\mathrm{op}} \to J \times J'$ and understand this as left-acting on the external product of families $f \times g : X \times Y \to J \times J'$, encode the span as a single structure map $u : Z \to X \times Y$, then define a biaction on the span to be a left action of $p \times q^{\mathrm{op}}$ on the family $(p\times q) \circ u$ which makes $u$ into a (fiberwise-)equivariant map. None of this really required that we work with groups instead of monoids.

The example that's motivating all of this is that if we have a (small, strict) category $\mathcal{C}$ with object set $J$ and a fibered group $p : G \to J$ then we can form a new category $\mathcal{C}'$ with object set $G$ equipped with an objectwise surjective equivalence of categories onto $\mathcal{C}$, essentially by replacing each object $j$ of $\mathcal{C}$ with the contractible groupoid whose object set is $p^{-1}(j)$ (the "universal bundle" groupoid of the "classifying space" groupoid $B (p^{-1}(j))$). This construction can also be understood as base change of the monad $\mathcal{C}$ in the double category of spans along the vertical arrow $p$. If we let $X = G$ with the obvious left $G$-action and $Y = G$ with the obvious right $G$-action then the span $X \leftarrow \operatorname{Mor}(\mathcal{C}') \rightarrow Y$ admits a $(G, G)$-biaction.

I would like to understand what I've outlined here in a cleaner way. I don't know if my definition of a biaction of a fibered group on a span is at all right, and I don't see a good "formal" construction of the biaction on the span in my motivating case. The specific family of groups and category I've been considering are the symmetric groups and the simplex category. Does anyone see a cleaner formalism for what I've written here, or recognize the ideas at all? I'd really appreciate a better high level pov in the hopes that it would make it clearer how or why the fibered biaction on the base change category arises.

Sorry for the wall of text, I'm happy to clarify things

David Michael Roberts (Nov 01 2023 at 04:51):

Note that a fibred group is a groupoid where the source and target maps coincide. Is the biaction of a fibred group the same thing as a profunctor from the groupoid to itself? Haven't checked, myself.

Brendan Murphy (Nov 01 2023 at 16:20):

Pretty much, it ignores the stuff about spans but that's okay. On the other hand I'm not sure how to get that category C' in terms of profunctors. If we base change along the functor BG -> discrete(J) then we don't get the fattened up version of C but instead a copy of C with the trivial action

Brendan Murphy (Nov 01 2023 at 16:22):

My C' category is a monad on EG in Prof, where EG is the disjoint union of the indiscrete groupoid on each fiber. In fact C' can be recovered as the base change of the map EG -> discrete(J) which is p on objects

Brendan Murphy (Nov 01 2023 at 18:38):

Maybe I could look at EG -> discrete(J) as an internal groupoid in Cat and consider internal profunctors on it?