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.

  • A question, more than a comment: why do you forget that Span is a bicategory? If you don't, you can rely on the "theory" (a rather euphemistic word :)) of categories enriched in a bicategory. – Fosco Loregian Aug 25 '17 at 17:08
  • I want to think about categories where the composition map is a span, not categories where the homsets are spans. The reason is that there are lots of interesting examples e.g. coming from the $S_\bullet$ construction that I'm interested in studying, and they're examples of categories enriched in the monoidal category of spans, not examples of categories enriched in the bicategory of spans. – Tim Campion Aug 25 '17 at 21:14

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.

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