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.

  • $\begingroup$ 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. $\endgroup$
    – fosco
    Aug 25, 2017 at 17:08
  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Aug 25, 2017 at 21:14

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy