Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?

Question

Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \mathrm{B}H \to \mathrm{B}G$ of (probably I should say "derived") stacks.

I would like to describe $\mathrm{B}G$ in terms of some sort of "groupoid" with base $\mathrm{B}H$. In some sense I know how to do this. If $f: N \to M$ were a surjective submersion of spaces, then $M$ would be equivalent to the "relative pair groupoid" of $N$, which you should try to build by taking the fibers of $f$ and taking the pair groupoid of each. I think you can write the nerve of this groupoid by saying that it has object space $M$, and space of 1-simplicies $M \times_N M$, and space of 2-morphisms $M \times_N M \times_N M$, and so one.

So I assume I can describe a stack equivalent to $\mathrm{B}G$ in terms of a simplicial object whose $k$-simplices are a $(k+1)$-fold homotopy-fibered-product, aka derived intersection, of $\mathrm{B}H$ with itself inside $\mathrm{B}G$.

But that's not very explicit, because I have no idea what derived intersections of classifying stacks of algebraic groups look like.

I'd much rather say something like "$\mathrm{B}G$ is the quotient groupoid for the action of SOMETHING on $\mathrm{B}H$". For comparison, if I have a surjection $H \to G$ of groups, then the underlying space $G$ is equivalent to the quotient groupoid of the space $H$ modulo the action of $\mathrm{ker}(H \to G)$ acting on $H$ by left (say) multiplication.

So my real hope is to do something with, say, the derived kernel of the inclusion $H \to G$. Not that I really know what those are in this nonabelian world. In my case, $H$ is not a normal subgroup of $G$, but nevertheless my hope-against-hope is to use some part of the cokernel of $H \to G$, which should have to do with the shift "$\mathrm{B}$".

How close are these hopes to reality?

Answer 1

Yes, there is something to this effect.

In fact there is a very general context for this. Since I know you are amenable to $\infty$-categories, I will use that language.

The homotopy theory of spaces is the initial example of an $\infty$-topos and one of the basic axioms of $\infty$-topoi is that "groupoids are effective". For spaces we also have that morphisms $f: U \to X$ which are surjective on $\pi_0$ are "effective epimorphisms". See for example sections 6.1 and 6.2 in Lurie's Higher Topos Theory.

If $f:U \to X$ is a map then we can form the Cech nerve $\infty$-groupoid $U_\bullet$. This is an $\infty$-groupoid which is internal in spaces. It is formed exactly like the pair groupoid you mentioned except that you use homotopy fiber products instead of fiber products and you do this for all finite fiber products to get a simplicial object (note that actually in the case where $f$ is a surjective submersion, and the spaces are reasonable these homotopy fiber products agree with ordinary fiber products).

What this means for spaces is that if $f: U \to X$ is surjective on $\pi_0$, then $X$ is the (homotopy) colimit of $U_\bullet$. The space of objects of $U_\bullet$ is $U$ and so this should be thought of as a generalization of an action on $U$. This says that $X$ is the (homotopy) colimit of this "action" on $U$. Although this "action" just means there is a groupoid extending $U$; the "groups" can change as you move around in $U$.

In your specific case things simplify a bit. You are looking at $f: BH \to BG$, which indeed is surjective on $\pi_0$. So this general machinery applies. Is says that there is an $\infty$-groupoid in spaces with object space $U_0 = BH$ whose homootpy quotient is $BG$. Moreover you can compute the hom spaces directly as iterated homotopy pull-backs. This "derived intersection" as you call it is not as mysterious as I think you make it out to be.

The first one is $$U_1 = BH \times_{BG}^h BH = |( G \; // \;H \times H)|, $$ the classifying space of the action groupoid of $H \times H$ acting on $G$ (by multiplication on both the right and left. The two projections to $BH$ should be clear from this description. The higher fiber products are similar but with $n$-copies of $G$ and $n+1$-copies of $H$. In particular they are all homotopy 1-types.