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I thought it was ridiculous that Span redirected to (infinity,n)-category of correspondences, so I made a stubby page for it instead.
Thanks. Added pointer to this page from the Examples-section at 2-category.
Good. I think there is a lot more cross-linking that could be done here, and a lot more that could be added, but I don’t have time right now.
Just saw this interesting paper on the arxiv that gives a universal property for the bicategory of polynomials: https://arxiv.org/abs/1806.10477v1 . The intro goes over a universal property from Hermida (“Representable Multicategories”) that really struck me as being about proarrow equipments so I thought I’d ask here.
It says that a pseudofunctor out of Span(E) (as a bicategory) is the same thing as a “Beck pseudofunctor” out of E, which is a pseudofunctor out of E where every morphism has a right adjoint and satisfies a Beck-Chevalley condition. But it seems to me like we could slightly strengthen this result and make it more natural by noting that Span(E) is naturally an equipment (assume E has pullbacks) and in any map of equipments $F : C \to D$ a vertical morphism of $C$ gets sent to a vertical morphism in $D$ and thus an adjoint pair of horizontal morphisms in $D$. Then the very natural (if you like equipments) theorem would be that any pseudofunctor $DblSpan(E) \to C$ of equipments is equivalent to a functor $E \to Vert(C)$ where $Vert(C)$ is the vertical category of $C$ where $DblSpan$ is the double category of spans rather than the bicategory.
Then we recover the original theorem by noting that the double category of adjunctions is a right adjoint, i.e., the maximal way to give a bicategory vertical arrows: a functor $C \to Adj(D)$ of equipments is the same as a pseudofunctor of bicategories $Hor(C) \to D$, where $Hor$ is the horizontal bicategory of $C$. So to sum it up it looks like we can decompose the functor $Span : CatPbk \to Bicat$ into two left adjoints:
$(DblSpan \dashv Vert) : CatPbk \to Equip$ $(Hor \dashv Adj) : Equip \to Bicat$so a pseudofunctor $Span(E) \to D$ is the same as $Hor(DblSpan(E)) \to D$ is the same as a functor $E \to Vert(Adj(D))$ which is the same as a Beck pseudofunctor.
So that seems very nice to me: Span is the minimal way to give a category horizontal arrows and Adj is the maximal way to give a bicategory vertical arrows. If that’s all true is there some way to make formulate Walker’s result about polynomials in terms of triple categories? Out of my depth there because I don’t actually know anything about triple categories but the formulation he uses involves a triple adjunction so maybe!
You seem to have dropped the crucial Beck-Chevalley condition?
You’re right, but it looks like dropping the Beck-Chevalley condition is exactly what leads to a certain generalization of the theorem that uses normal oplax functors rather than pseudofunctors [1], which sounds even more like equipment stuff to me. I’ll come back to this later because I am currently using it to procrastinate before a deadline :)
[1] Dawson, Pare and Pronk, Universal Properties of Spans
Note that a sequel to that DDP paper, The span construction, uses double categories explicitly.
Ah perfect, that’s exactly what I was looking for: they go all the way and define Span to be an “oplax normal double category with all companions and conjoints” which might be also named a co-virtual equipment.
So you get that Span gives you a co-virtual equipment and that it is the free co-virtual equipment wrt the forgetful functor that takes the vertical category and it is representable iff the original category has pullbacks.
Would be very cool if polynomials turn out to be some kind of free virtualized triple category that’s representable when the category is locally cartesian closed.
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