Configuration topos?

Question

Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space $C_k(X)$ of $k$ distinct points in $X$. It also contains topological information about how these configuration spaces fit together, which is what I'm interested in. For example, if $X$ is a finite topological space (e.g. the "4-point circle") then ${\bf Fin}/X$ has the structure of a topos.

Q1: In the finite $X$ case, how should the topos ${\bf Fin}/X$ be considered topologically? Can you point me to some references?

Q2: If $X$ is not finite, what is the proper way to think of ${\bf Fin}/X$? For example, surely ${\bf Fin}/{\mathbb R}^n$ has been well studied -- what are good references? What are the topological properties of various adjunctions, say ${\bf Fin}/X\to{\bf Fin}/Y$, given a map $Y\to X$.

Q3: What if $C$ is a finite category and, throughout the discussion above, we replace ${\bf Fin}$ with the topos $C-{\bf Fin}$ of functors $C\to{\bf Fin}$. If ${\bf Top}$ is the category of topological spaces and $X\colon C\to{\bf Top}$ is a functor, we again can discuss $C-{\bf Fin}/X$. Any references for this?

Q4: Instead of references, can you think of anything interesting to say on this topic -- something that you think may inform me, inspire me, or lead me to ask better questions? Please consider me totally ignorant so that no amount of baby talk could possibly insult me.

Thanks!

Answer 1

If $X$ is finite, then as a mere category $\mathbf{Fin}_{/ X}$ is equivalent to the cartesian power $\mathbf{Fin}^X$. If $X$ is infinite, then $\mathbf{Fin}_{/ X}$ should be thought of as the full subcategory of $\mathbf{Set}^X$ spanned by the $\aleph_0$-compact objects. This is not a particularly nice category – for example it does not have a terminal object. But I suspect what you are interested in has nothing to do with $\mathbf{Fin}_{/ X}$ qua category but rather as some kind of generalised orbifold. So let us topologise it.

Let $\mathbb{C}$ be an internal category in a category $\mathcal{E}$ with finite limits. Then we can speak of internal presheaves on $\mathbb{C}$ (i.e. "internal" functors $\mathbb{C}^\mathrm{op} \to \mathcal{E}$): such a thing consists of an object $F$, a morphism $p : F \to \operatorname{ob} \mathbb{C}$, and a right action of $\mathbb{C}$ on $F$ (i.e. a morphism $F \times_{\operatorname{ob} \mathbb{C}} \operatorname{mor} \mathbb{C} \to F$ making various diagrams commute; note the pullback is of $\operatorname{codom} : \operatorname{mor} \mathbb{C} \to \operatorname{ob} \mathbb{C}$, not $\operatorname{dom}$). One can then form another internal category, say $\mathbb{F}$, where $\operatorname{ob} \mathbb{F} = F$ and $\operatorname{mor} \mathbb{F} = F \times_{\operatorname{ob} \mathbb{C}} \operatorname{mor} \mathbb{C}$, and there is an internal "discrete" fibration $\mathbb{F} \to \mathbb{C}$. (Here "discrete" is meant in the categorical sense, not topological!)

Now, when $\mathcal{E} = \mathbf{Top}$ and $\mathbb{C}$ is topologically discrete, an internal presheaf on $\mathbb{C}$ is the same thing as a functor $\mathbb{C}^\mathrm{op} \to \mathbf{Top}$. In particular we can do this for $\mathbb{C}$ the category of finite cardinals and $F$ the presheaf defined by $F (n) = X^n$. The ordinary category corresponding to the internal category $\mathbb{F}$ is easily seen to be the comma category $(\mathbb{C} \downarrow X)$, so we have achieved the desired topologisation.

More explicitly, $\operatorname{ob} \mathbb{F}$ is the disjoint union $\coprod_{n \in \mathbb{N}} X^n$, and $\operatorname{mor} \mathbb{F}$ is the disjoint union $\coprod_{f : m \to n} X^n$. Since $\operatorname{codom} : \operatorname{mor} \mathbb{C} \to \operatorname{ob} \mathbb{C}$ is a local homeomorphism, the same is true for $\operatorname{codom} : \operatorname{mor} \mathbb{F} \to \operatorname{ob} \mathbb{F}$. Thus, if we restrict to the maximal subgroupoid $\mathbb{G} \subseteq \mathbb{F}$, we have a so-called étale topological groupoid. (Strictly speaking we must check that inversion is a homeomorphism, but we could just run through the above arguments replacing $\mathbb{C}$ with its maximal subgroupoid.) In the case where $X$ is sober and locally compact, we can equally regard $\mathbb{G}$ as an étale localic groupoid. Either way, we can define the topos $\mathbf{Sh}(\mathbb{G})$ of equivariant sheaves on $\mathbb{G}$ (= internal presheaves $P$ on $\mathbb{G}$ where the map $P \to \operatorname{ob} \mathbb{G}$ is a local homeomorphism) – the so-called classifying topos of $\mathbb{G}$. It should go without saying that the orbit space of $\mathbb{G}$ is just the disjoint union of the configuration spaces of $X$.

One could also ask about sheaves on $\mathbb{F}$, but the subject of sheaves on topological categories seems to be less well-studied. It is the case that $\mathbf{Sh}(\mathbb{F})$ is a topos. One reference for this is [Moerdijk, Classifying spaces and classifying topoi].

Answer 2

I still don't understand the formulation of the question, so let me make a brief comment about fitting together configuration spaces. One way to think about $C_k(X)$ is as the space of injective maps $[k] \to X$. Since injective maps compose, this means that any injective map $[i] \to [j]$ induces a map $C_j(X) \to C_i(X)$ on configuration spaces. This makes the collection of configuration spaces $C_i(X)$ into an $\text{FI}^{op}$-space in the sense of Church, Ellenberg, and Farb, namely an element of the category of functors $\text{FI}^{op} \to \text{Top}$ where $\text{FI}$ is the category of finite sets and injections. This is an enhancement of considering the natural action of the symmetric groups on each configuration space and it turns out that this extra structure can be used to prove or at least reprove a lot of nice results, e.g. about the limiting behavior of the cohomology of configuration spaces (which naturally becomes an $\text{FI}$-module, defined similarly).

Answer 3

I think my question, at least the one I meant to ask, is probably best answered by a reference to Michael Weiss's papers on embedding calculus. Especially the Grothendieck topologies $J_k$ seem to be of interest.

Answer 4

If $X$ is a topological space:

One object you might be contemplating is simply the quotient stack $[X^n/\Sigma_n]$ (where $\Sigma_n$ is the symmetric group). It is a étale stack, and so one can conside sheaves on its étale site. Those form a topos.

You can also consider he category whose objects are families of sheaves $(\mathcal F_n)$, where each $\mathcal F_n$ is a sheaf over $X^n=\mathrm{Map}(\{1,\ldots,n\},X)$, equipped with a morphism of sheaves from $\mathcal F_n$ to $(-\circ f)_*\mathcal F_m$ for any map $f:\{1,\ldots,n\}\to\{1,\ldots,m\}$ (those maps are subject to some obvious associativity condition). Those also form a topos. This is probably what you call $\mathbf{Fin}/X$.

To your question "should the topos $\mathbf{Fin}/X$ be considered topologically?" I would say "yes". But it's not the topos of sheaves on some topological space... and it's also not the topos of sheaves on some topological stack. So I'm not sure what exactly you want to know about it.