The category theory of Span-enriched categories / 2-Segal spaces

Question

The category $\mathsf{Span}$ of spans of sets is symmetric monoidal closed under $\times$ (the cartesian product from $\mathsf{Set}$, which is not the categorical product in $\mathsf{Span}$), complete and cocomplete. So it should be a nice category to enrich in! A $\mathsf{Span}$-enriched category is "a category with multivalued composition".

Caveat: Actually, one issue is that $\mathsf{Span}$ is naturally a (2,1)-category, but I think this complication can be dealt with.

Motivation: In fact, if we $\infty$-ify and look at $\infty$-categories enriched in spans of spaces, we should get something like the 2-Segal spaces of Dyckerhoff and Kapranov i.e. the decomposition spaces of Galvez-Carillo and Kock and Tonks, which are surprisingly common, arising e.g. from the $S_\bullet$ construction of algebraic $K$-theory. Without $\infty$-fying, a $\mathsf{Span}$-enriched category is essentially a unital 2-Segal set, which is a simplicial set satisfying certain Segal conditions. For the purposes of this question I'm happy to stick to the non - $\infty$-context, but that's where I'd ultimately like to go with this.

The problem: However, it seems that the usual notions of enriched category theory shouldn't just be applied blindly to $\mathsf{Span}$-enriched categories. For example:

• A $\mathsf{Span}$-enriched functor $F: C \to D$ will have spans $F_{c,c'}$ from $C(c,c')$ to $D(Fc,Fc')$, but for many purposes, one will be interested in enriched functors $F$ such that $F_{c,c'}$ is an actual function. Call such an enriched functor map-like. For example, the simplicial maps between 2-Segal sets correspond to the map-like enriched functors.

• As one illustration of this, note that modulo size issues, the inclusion $I$ of ordinary categories into $\mathsf{Span}$-enriched categories has a right adjoint, given by the usual "underlying category" functor, which in this case sends $C$ to what Dyckerhoff and Kapranov call the "Hall category" $HC$ of $C$, where $HC(c,c') = \mathsf{Set}^{C(c,c')}$ and composition given by an indexed coproduct. The inclusion $I$ also has a left adjoint $F$, which identifies all the different composites of any two morphisms -- but $F$ is only functorial with respect to map-like enriched functors! (Simplicially, this is the Bousfield localization turning a 2-Segal set into a 1-Segal set, i.e. a category.) In fact, if the morphisms of $\mathsf{Span}$-enriched categories are taken to be all enriched functors, then $I$ fails to preserve products and so has no left adjoint.

• Enriched (co)limits are probably not the correct notion of (co)limits for $\mathsf{Span}$-enriched categories. For example, if $C$ is an ordinary category with (co)products, then the corresponding $\mathsf{Span}$-enriched category $IC$ will typically not have (co)products. This comes down to the fact that (co)limits in a $\mathcal{V}$-enriched category are defined in terms of limits in $\mathcal{V}$, and limits in $\mathsf{Span}$ are not at all related to limts in $\mathsf{Set}$.

My question: Has the category theory of $\mathsf{Span}$-enriched categories -- notions of functor, (co)limit, etc -- been developed somewhere in the literature? If it hasn't been developed specifically, is there some existing formal-category-theoretic framework that it should fit into? Equipments, $\mathcal{F}$-categories,... the ones I know of don't seem to quite fit the bill.

Answer 1

A solution to achieving the kind of enrichment you are looking for (though I cannot say what precise comparison it would give with $2$-Segal spaces) may be to actually remember not only the $2$-categorical structure but also the double categorical one. While the standard notion of enrichment over an equipment suggested by Fosco's comment does indeed not give the sought-after result, it can still pay to view spans as an equipment $\mathbb{Span}$, and in fact a monoidal equipment (which they do form, cf. Hansen–Shulman's "Constructing symmetric monoidal bicategories functorially", Example 2.11 and unpacking after Definition 2.10, and section 3 for the compatibility with companions and conjoints).

If $(\mathbb{V},\otimes,1)$ is a monoidal proarrow equipment, one can (sketchily, cf. remark at the end) define a $\mathbb{V}$-enriched category $C$ to consist of a collection of objects, hom-objects $C(c,c^{\prime})$ of $\mathbb{V}$ between pairs of objects, composition proarrows $C(c^{\prime},c^{\prime\prime})\otimes C(c,c^{\prime})⇸ C(c,c^{\prime\prime})$ and unit proarrows (we also want unit spans rather than functions) $1⇸ C(c,c)$, structured by associativity and unitality constraint cells (probably to require invertible). Note that so far we have not used the vertical part of $\mathbb{V}$ (and in fact the equipment property was just to reassure us that we have not lost much by disregarding these vertical arrows).

Where the double categorical structure becomes really helpful is in defining the $\mathbb{V}$-enriched functors of $\mathbb{V}$-categories: we will say that a (vertically) $\mathbb{V}$-enriched functor $F\colon C\to D$ assigns (to an object $c\in C$ an $Fc\in D$ and) to every pair $(c,c^{\prime})$ of objects a vertical arrow $C(c,c^{\prime})\to D(Fc,Fc^{\prime})$, along with functoriality cells $$ \begin{array}{ccc} C(c^{\prime},c^{\prime\prime})\otimes C(c,c^{\prime}) & ⇸ & C(c,c^{\prime\prime}) \\ \downarrow & \Downarrow & \downarrow \\ D(Fc^{\prime},Fc^{\prime\prime})\otimes D(Fc,Fc^{\prime}) & ⇸ & D(Fc,Fc^{\prime\prime}) \end{array}\qquad\text{ and }\qquad \begin{array}{ccc} 1 & ⇸ & C(c,c) \\ \downarrow & \Downarrow & \downarrow \\ 1 & ⇸ & D(Fc,Fc) \end{array} $$ in $\mathbb{V}$ (which you may want to require invertible). In the case of $\mathbb{V}=\mathbb{Span}$, this gives exactly the kind of map-like functors you were seeking. You should also be able to similarly define $\mathbb{V}$-natural transformations (the more natural definition seems to be again a vertical one).

If you want to do more category theory (so have well-behaved notions of limits, etc. built in the theory), there remains the question of constructing a proarrow equipment of $\mathbb{V}$-enriched categories. If $\mathbb{V}$ is monoidal closed, so that it can be self-enriched, the previous construction should be enough to give a $\mathbb{V}$-presheaf-style definition of $\mathbb{V}$-profunctors — or more abstractly, since we have given they data for a $2$-category of $\mathbb{V}$-categories, you should recover them from codiscrete two-sided cofibrations therein. For example, $\mathbb{Span}$ is monoidal closed, with internal hom from $X$ to $Y$ given by $X\times Y$ and composition spans $X\times Y\times Y\times Z⇸ X\times Z$ given by the evaluation $Y\times Y\leftarrow Y\to1$ (since pullbacks over $Y$ can be computed as basechanges of the diagonal $\Delta_{Y}$, this does enrich the usual composition of spans).

Note: The kind of enrichment defined above will probably feel a bit wobbly in practice, as a result of forcing a $2$- or in fact double categorical structure into a $1$-categorical one. Indeed, as Leinster explains in "Generalized Enrichment for Categories and Multicategories", the most general structure we expect to sensibly enrich categories in is virtual double categories, but I think of monoidal (virtual) double categories as closer in spirit to monoidal $2$-categories and thus adding a categorical level in an orthogonal direction.

Answer 2

You might like to look at the paper "Algebraic theories, span diagrams and commutative monoids in homotopy theory" (https://arxiv.org/abs/1109.1598) by James Cranch. I think that it does not directly answer any of your questions, but it involves the same circle of ideas.