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Stream: learning: questions

Topic: The double category of spans

John Baez (Aug 27 2025 at 13:56):

Dawson, Pare and Pronk have the following nice characterization of the double category $\mathbb{C}\mathbf{sp}(\mathsf{A})$ of cospans in a category $\mathsf{A}$ with pushouts. Roughly it says that $\mathbb{C}\mathbf{sp}(\mathsf{A})$ is the free fibrant double category on $\mathsf{A}$:

Theorem 3.15. Let $\mathsf{A}$ be a category with pushouts and $\mathbb{D}$ a fibrant double category. Then composing with the inclusion $\mathsf{A} \to \mathbb{C}\mathbf{sp}(\mathsf{A})$ gives an equivalence between the category of normal lax double functors from $\mathbb{C}\mathbf{sp}(\mathsf{A})$ to $\mathbb{D}$ and the category of functors from $\mathsf{A}$ to $\mathbb{D}_0$ , the category of objects and tight morphisms in $\mathbb{D}$ .

John Baez (Aug 27 2025 at 13:58):

However I'm wondering which functors from $\mathsf{A}$ to $\mathbb{D}_0$ correspond to ordinary (that is, pseudo) double functors from $\mathbb{C}\mathbf{sp}(\mathsf{A})$ to $\mathbb{D}$, rather than merely normal lax ones.

James Deikun (Aug 27 2025 at 14:56):

It's pushout-preserving functors. You can see this by looking at the two obvious ways you can write the cospan in a pushout square as a composite of a conjoint and a companion (in different orders) in $\mathbf{\mathbb Csp}(\mathsf A)$.

John Baez (Aug 27 2025 at 15:08):

Wow, nice. Pushout preservation is clearly sufficient, so I guess you're suggesting I look at that stuff to prove the necessity. Thanks!

James Deikun (Aug 27 2025 at 15:20):

Yeah, the universal property of the composite, knowing that it's of the form $X(f,g)$, implies the universal property of a pushout for the cospan

$$\bullet \xrightarrow{f} X \xleftarrow{g} \bullet$$

John Baez (Aug 27 2025 at 15:29):

Okay, I got it. Thanks.

Nathanael Arkor (Aug 27 2025 at 16:14):

The result James mentions appears as Theorem 3.24 in the paper you linked to.

John Baez (Aug 27 2025 at 16:32):

Even better! I looked for it, after he mentioned it, but somehow missed it. :blushing:

John Baez (Aug 27 2025 at 16:57):

Oh, now I see why I missed this result. It seems they prove it only when $\mathbb{D}$ obeys some sort of Beck-Chevalley property. Is that just a special case of what I was asking about?

(They also demand that $\mathbb{D}$ be fibrant and that its underlying category of tight morphisms has pushouts, but all that is fine.)

(And again, I'm taking their results and dualizing them.)

Kevin Carlson (Aug 27 2025 at 17:18):

Since $\mathbb{C}\mathbf{sp}(\mathsf{A})$ satisfies Beck-Chevalley and double functors preserve companions and conjoints, if you have a pushout-preserving functor $F:\mathsf{A}\to \mathbb{D}_0$, then for there to exist any extension to a pseudo-double functor $\bar F:\mathbb{C}\mathbf{sp}(\mathsf{A})\to \mathbb{D},$ we can see $F$ will have to send every pushout square in $\mathsf{A}$ to a Beck-Chevalley square in $\mathbb{D}.$ So that shows the Beck-Chevalley condition is essentially necessary, if you're going to start with a pushout-preserving functor on $\mathsf{A}.$ For that matter, even if you don't start by assuming $F$ preserves pushouts, it will still have to send pushout squares to some-kind-of-square-with-the-Beck-Chevalley-property!

Kevin Carlson (Aug 27 2025 at 17:20):

So perhaps there's a generalization of 3.24 where you decorate $\mathbb{D}_0$ with the squares you know are Beck-Chevalley in $\mathbb{D}$, and then pseudo-functors $\mathbb{C}\mathbf{sp}(\mathsf{A})\to \mathbb{D}$ correspond to arbitrary functors $\mathsf{A}\to\mathbb{D}_0$ which send pushout squares to Beck-Chevalley squares. But that seems like so little gain over the theorem as stated, in particular by having to add extra structure to $\mathbb{D}_0$, a structure which is hard to figure out if the Beck-Chevalley squares aren't just the pushouts, I think you can see why they didn't bother.

John Baez (Aug 27 2025 at 17:57):

Thanks - I'm thinking about this. I was initially put off by the rather complex diagram in their definition of 'Beck-Chevalley double category' in Definition 3.22. But I'm reassured by how you're talking about it as if it's just the usual Beck-Chevalley stuff I know and (don't quite) love. Now I'm daring to actually look at that diagram and see that yeah, it's the Beck-Chevalley condition.

John Baez (Aug 27 2025 at 18:10):

I may never have thought about fibrant double categories satisfying the Beck-Chevalley condition. I guess all structured or decorated cospan categories obey it!

Kevin Carlson (Aug 27 2025 at 18:26):

I agree for structured cospans, at least! I don't have much on the top of my head about companions and things in double categories of decorated cospans.

Patrick Nicodemus (Aug 27 2025 at 18:39):

I have been spending a lot of time recently writing up a proof of this, I didn't realize it was already proven. I'm disappointed to have spent so much time on something unnecessary.

On the other hand I have not gone through this paper in detail and I suspect that my write up establishes some useful categorical generalities for 2-dimensional algebra that go beyond this paper and would be helpful in proving similar theorems.

Patrick Nicodemus (Aug 27 2025 at 18:44):

Damn. This is not the first time I have spent weeks or months on a theorem like this for nothing. Not that the proof of the theorem itself took months, but many categorical generalities established basically for the sake of situating the theorem in a proper context of general 2-dimensional categorical algebra that I would not have bothered with at all if I knew this theorem was already proven.

Patrick Nicodemus (Aug 27 2025 at 18:49):

I am interested in spans not cospans. @John Baez maybe we could share our results here as I need the dualized result about spans as well rather than cospans.

John Baez (Aug 27 2025 at 20:20):

@Patrick Nicodemus - I'm sorry to hear you were proving such a theorem. I found out about it from @Nathanael Arkor here on August 11th.

I haven't actually proved any results like this; I'm trying to use these results to study structured and decorated cospan categories.

Patrick Nicodemus (Aug 27 2025 at 20:21):

I haven't gone through the paper yet so I may be over reacting, but it seems that one should blabber constantly about what they're working on in order to prevent situations like this.

John Baez (Aug 27 2025 at 20:22):

Yes, that's one reason I blabber constantly about what I'm working on.

Patrick Nicodemus (Aug 27 2025 at 20:25):

For the record, my characterization is a purely 2-categorical categorization of what I call the pseudocategory of spans, which need not be internal to the 2-category Cat. If A is an object in the 2-category and C is a pseudocategory internal to the 2-category, I can also characterize lax functors C -> Span(A) by reference to only C and A.

Mike Shulman (Aug 27 2025 at 20:25):

Public "blabbering" about one's work is one approach, but many people seem to do okay without it. I think more common approaches are to ask around about what's already known, either publicly or privately or both, and do research on mathscinet/arxiv/etc., following links and references.

John Baez (Aug 27 2025 at 20:29):

Yes, I also ask a lot of questions, like the one that started this thread. And I didn't really say much about what I'm actually working on, which is why I'm interested in this issue.

Patrick Nicodemus (Aug 27 2025 at 20:43):

Well, now I feel like I should say what I'm currently thinking about just to see if anyone has pointers. If nothing else it seems like this is a good opportunity to discuss the matter because John is obviously interested in this double category of spans as well.

It should be the case that lax functors $C \to\operatorname{Span}(A)$ can be described in terms of arrows $\operatorname{Ob}(C) \to A, \operatorname{Arr}(C)\to A$ and various compatible 2-cells between them, which seems to be concisely describable by saying that the distinguished maps $\operatorname{Ob}(C) \to A, \operatorname{Arr}(C)\to A$ are themselves the Ob and Arr of a pseudocategory whose structure is compatible with that of $C$. A bit more formally, if $\mathcal{E}$ is the ambient 2-category then one can form the 2-category $\mathcal{E}/A$ which is fibered over $\mathcal{E}$ by a projection functor $\pi : \mathcal{E}\to\mathcal{A}$, and then the idea is that lax functors $C\to \operatorname{Span}(A)$ should correspond to pseudocategories internal to $\mathcal{E}/A$ which lie strictly over $\mathcal{C}$ via $\pi$. Then, if the pseudocategory of spans actually exists, then the identity functor $\operatorname{Span}(A)\to \operatorname{Span}(A)$ should itself correspond to a pseudocategory structure in $\mathcal{E}/A$ lying over $\operatorname{Span}(A)$, and the pair of these objects should play some kind of special role in the fibration.

The main technical obstacle I have to work out is: $\pi:\mathcal{E}/A\to\mathcal{E}$ is a 2-categorical fibration in some sense, and we need to talk about pullbacks in $\mathcal{E}/A$ to make sense of a pseudocategory in this category, do we have limit lifting theorems for 2-fibrations that assure us that pullbacks are computed in the "obvious" way in $\mathcal{E}/A$? More generally how do we compute (perhaps weighted) limits in $\mathcal{E}/A$ relative to $\mathcal{E}$? So I am thinking that when studying representability in a 2-category, combined with pseudocategories and other concepts of 2-categorical algebra involving (probably finite) limits such as 2-Lawvere theories or 2-sketches, we are eventually going to want to do 2-categorical algebra in the total category arising from a 2-presheaf and there we will need a (weighted) limit lifting theorem for 2-fibrations.

Patrick Nicodemus (Aug 27 2025 at 20:45):

So for a while I have been writing up limit lifting theorems for fibrations between 2-categories and bicategories with applications to this object of spans as my primary motivation.

Patrick Nicodemus (Aug 27 2025 at 20:52):

This is why I asked Nathanael about this here #theory: category theory > Fibrations and limits @ 💬

Nathanael Arkor (Aug 28 2025 at 07:33):

[DPS10] do not consider Span as a right adjoint; only as a left adjoint. Their universal property is therefore of a different nature to yours. The only place I am aware of in which Span is exhibited as a right adjoint (in the 2-categorical literature) is Vaugelade's 3-page note Un adjoint au foncteur Span. She establishes that:
$\text{Cat}_{pb}(T\sigma L(\mathscr B), \mathcal E) \cong \text{Bicat}_l(\mathscr B, \mathbf{Span}(\mathcal E))$
for some functor $T\sigma L$ (here $L$ is the lax morphism classifier for bicategories, and $T\sigma$ is a construction arising from the theory of limit sketches).

Nathanael Arkor (Aug 28 2025 at 07:44):

So I would not be deterred from continuing your investigation (though it would be worth taking a look at Vaugelade's two papers to see whether there is anything to be learnt from her approach).

Nathanael Arkor (Aug 28 2025 at 07:46):

(I would be surprised if no-one had described limits in slice 2-categories in the literature, though I haven't tried looking.)

Patrick Nicodemus (Aug 28 2025 at 18:24):

Interesting. Thank you Nathanael, I will take a look at this.