Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: opcartesian vs U.P. of composites

Noah Chrein (Oct 25 2025 at 23:14):

Hello!

tl;dr: why do we call opcartesian cells composites when the opcartesian property does not reflect the universal properties of the composites in the examples?

For context, I have been working on developing a theory of virtual double quasi-categories, and virtual double quasi-equipments as part of a program to develop a notion of higher virtual double equipment analogous to $\Theta_n-$ spaces. In this pursuit I am naturally asking the question what is the correct generalization of op/cartesian cells. For formal properties, i.e. doing all the companion cell stuff, it seems like the correct generalization is straight forward (we require a 2-simplex witnessing the classical factorization property unique up to equivalence).

However, I began to wonder why exactly we call the codomains of opcartesian cells composites, as in SC10 section 5. Well, clearly in the examples span and prof, the opcartesian cells are the ones whose codomains are defined by pullback and coend, which in the classical pseudo double category setting are the horizontal composites, this seems reasonable.

However, if you dig into what opcartesian implies in the example of span(c), it isn't actually equivalent to the universal property of the pullback, rather, it is equivalent to the fact that the domain span is isomorphic to the codomain span. Of course, this comes from the fact that the pullback is used in the very definition of cell in span(c). Put simply: The definition of cell involves using the correct notion of composite, thus the composite isn't really derived in some universal sense. This seems very tautological.

Now, I know that virtual composites + opcartesian implies horizontal composites. This is theorem 5.4 in SC10, but this doesn't hold in the generalization to virtual double simplicial sets, that is, we don't get a simplicial quasi-category or anything like that (we don't gain a notion of horizontal composite from opcartesian) this is because the virtual composition was necessary in theorem 5.4. That is

virtual composite + opcartesian = horizontal composites

but

opcartesian != horizontal composites (in the absence of virtual composites)

There's much more to this story that involves some definitions of these vdsSets, but I'll cut to a more basic question. I think the issue at the heart of this is that opcartesian doesn't reflect the universal properties of span(c) or v-prof. So my question is: why?

why do we call opcartesian cells composites when the opcartesian property does not reflect the universal properties of the composites in the examples?

Thank you

James Deikun (Oct 26 2025 at 01:43):

The universal property of opcartesian cells is an abstraction of the universal properties of the composites in particular examples. The composite of spans and the composite of profunctors and the composite of matrices don't have any universal property in common without considering what cells out of them look like. This is less tautological than it looks in general, though. The example of cospans is instructive. There a cell can be defined as a commutative triangulation of its boundary, and the universal property of a pushout falls out of that automatically. The VDC of cospans also, not coincidentally, has a very nice universal property as a VDC.

Noah Chrein (Oct 26 2025 at 14:12):

Thank you for your response, but the factorization property defining opcartesian cells does not reduce to the universal property of a pullback of spans $p_1 \times_{A_1} p_2$ for spans $A_i\leftarrow p_{i+1}\rightarrow A_{i+1}$. It reduces to the property that the cell $p_1 \times_{A_1} p_2\overset{\eta}\to q$ is an isomorphism.

James Deikun (Oct 26 2025 at 14:29):

The property of being isomorphic to a pullback of spans is precisely the same property as being a pullback of spans.

Noah Chrein (Oct 26 2025 at 16:49):

Yes there is a nuance here that I will try to communicate in more detail later, but in classic zulip fashion I figured out my own question the day after posting. Essentially if you replace the pullback $P = p_1\times_{A_1} p_2\overset \eta \to Q$ in the definition of a 1-cell in the span(C):vdc with an arbitrary object $P\overset \eta \to Q$ (with maps $P\to p_i$) the opcartesian property remains equivalent to $\eta$ being an isomorphism $P \overset{\eta}\cong Q$, but does not immediately imply that $P\cong p_1\times_{A_1} p_2$, rather it is baked into the definition of 1-cell in span(C). What I would like to see is a lemma for this weakened version of span(C) along the lines of:
"span(C) has opcartesian cells $\iff$ C has pullbacks"

Noah Chrein (Oct 26 2025 at 16:56):

The answer comes from how one chooses to weaken the notion of
$\text{span}:Cat_{pull}\to vdc$ to $\text{span}:Cat \to vdsSet$,
one generalization gives the universal property of pullbacks, the other doesn't (giving a good reason to favor one choice). If I finish writing it up I'll post later.

Nathanael Arkor (Oct 26 2025 at 19:08):

Span naturally forms a co-virtual double category, rather than a virtual double category, and then it is the case that Span admits composites iff the underlying category admits pullbacks. See Dawson–Paré–Pronk's "The Span Construction".

Noah Chrein (Oct 26 2025 at 19:43):

thank you Nathanael, this makes some sense to obtain the universal property. However, the point of a virtual double category instead of a co-virtual double category is to define modules, whence a module of spans is a category, I expect the module condition in virtual double quasi categories to reflect the segal condition written in terms of mapping spaces. Furthermore virtual equipments (as opposed to co-virtual equipments) are used to describe many formal category theory constructions in Riehl and Verity's $\infty$-cosmoi. I would like to retain these formal category theory capabilities while also capturing the universal property of composites in the examples.

but maybe getting the best of both worlds is why @Mike Shulman has been talking about polycategories?

Mike Shulman (Oct 27 2025 at 00:17):

I don't think what you want is possible. There are two kinds of universal properties, "mapping in" and "mapping out". Opcartesian cells in a vdc, or more simply in a multicategory, have a mapping-out univeral property. That makes them convenient for describing examples where the composites would naturally have a mapping-out universal property, but not for describing examples where the composites would naturally have a mapping-in universal property. It's just like there are two kinds of Set-valued functors, covariant and contravariant, and therefore two notions of representability.

Mike Shulman (Oct 27 2025 at 00:21):

There is a notion of double category where the cells have strings of arrows as both their domain and codomain. It's more like a prop than like a polycategory. If you do that for both the vertical and horizontal arrows, then @Aaron David Fairbanks and I called it an implicit double category. You might be able to get some mileage out of that, but I don't know whether you could put both modules with non-representable composites and spans with non-representable composites into a common framework that would actually be useful.

Noah Chrein (Oct 27 2025 at 01:30):

Thank you wizard Shulman. I will try a bit more, as I think there's a way to do it by defining the higher simpleces of cells in span(C) correctly, essentially allowing you to recover the mapping in property. The alternative is to add an extra condition (mimicking the U.P.) and define composites = opcartesian + extra condition. But I will look into this implicit double cat as well, I appreciate the pointer.

Noah Chrein (Oct 27 2025 at 01:34):

I guess I just had a problem with everyone calling opcartesian cells "composites" when the nature of them looking like composites is baked into the definition of the cells themselves, especially since the vdc + opcart = pdc theorem is lost in the generalization to vdsSets

Mike Shulman (Oct 27 2025 at 04:28):

The justification of the name "composite" is that from the perspective of an abstract double category, they are the only notion of "composite" available. In particular examples, it is of course the job of whoever is constructing the example to do it in such a way that they match the desired notion of "composite" for that example. Then it happens that sometimes the universal property of the desired "composites" can be "built into the opcartesianness", and sometimes it can't and has to be built into the definition of "cell".

Noah Chrein (Oct 29 2025 at 15:05):

Ah okay, I would like it to be a bit tighter otherwise the generalization of Span(C) to virtual double quasi categories fails. Seems to me either

composite = opcartesian + something else (perhaps the data of a choice of universal property)

or changing the virtual double part all together like you suggested earlier with IDCs.

James Deikun (Oct 29 2025 at 15:17):

Might I suggest that maybe if you can't seem to define a virtual double quasi-category of spans, the problem might be with your precise definition of virtual double quasi-category rather than with the concept itself? What are you defining a virtual double quasi-category as?

Nathanael Arkor (Oct 29 2025 at 15:24):

I was under the impression there already was an (infinity, 1)-categorical notion of virtual double category: namely Gepner and Haugseng's "generalized non-symmetric ∞-operads".

James Deikun (Oct 29 2025 at 16:20):

That's here?

Noah Chrein (Oct 29 2025 at 17:59):

James Deikun said:

Might I suggest that maybe if you can't seem to define a virtual double quasi-category of spans, the problem might be with your precise definition of virtual double quasi-category rather than with the concept itself? What are you defining a virtual double quasi-category as?

the problem isn't with virtual double quasi-category, that's defined, its with choosing the proper generalization of span(C) so that the opcartesian cells are the pullback

Noah Chrein (Oct 29 2025 at 18:00):

there's a walking generalized multicategory, its functors into a monad are T-simplicial sets, and (in certain cases including T = fc) there's a natural notion of segal fibration