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: loose exponentials

Matteo Capucci (he/him) (Aug 17 2023 at 21:53):

I have a monoidal double category $\mathbf D$ whose loose arrows 'form' a closed bicategory. That is, loose arrows $A \to B \otimes C$ are in bijection with loose arrows $A \to [B, C]$ for a certain object $[B, C]$ which I call the 'loose exponential'. Crucially, this is not a closed structure for the monoidal category of tight arrows.

The chief examples are Rel and Prof, where one actually has a coclosed structure too. Notice $[B, C] = B \times C$ in Rel, so you can see this is quite different from $C^B$, the 'tight exponential'.

I'm not completely sure how to express this structure, in particular what is exactly the universal property (I said above there is a bijection, but I suspect one could improve this to take into account tight morphisms as well). Before spending time fiddling with it I figured someone here might know something. Does anybody know if 'loose exponentials' have been studied before?

Mike Shulman (Aug 17 2023 at 22:09):

I would say it's just a bicategorical exponential in the bicategory of loose arrows.

Nathanael Arkor (Aug 17 2023 at 22:10):

Susan Niefield gave a talk about cartesian closed double categories at CT this year, and two of the examples were relations, and profunctors. Perhaps this is the sort of the universal property you're looking for?

Mike Shulman (Aug 17 2023 at 22:10):

Note that in Rel and Prof it's not just a closed monoidal structure but a compact closed structure.

Nathanael Arkor (Aug 17 2023 at 22:11):

Matteo Capucci (he/him) said:

I have a monoidal double category $\mathbf D$ whose loose arrows 'form' a closed bicategory.

(One thing to be careful about is that a "monoidal closed bicategory" is a different thing from a [[closed bicategory]]. (There's some awkwardness in the terminology here, because this is also different from the bicategorical analogue of a [[closed category]].)

Matteo Capucci (he/him) (Aug 18 2023 at 14:47):

Nathanael Arkor said:

Matteo Capucci (he/him) said:

I have a monoidal double category $\mathbf D$ whose loose arrows 'form' a closed bicategory.

(One thing to be careful about is that a "monoidal closed bicategory" is a different thing from a [[closed bicategory]]. (There's some awkwardness in the terminology here, because this is also different from the bicategorical analogue of a [[closed category]].)

Oh jeez! I wasn't aware of that. Yeah it's the monoidal closed kind I'm talking about.

Matteo Capucci (he/him) (Aug 18 2023 at 14:50):

Nathanael Arkor said:

Susan Niefield gave a talk about cartesian closed double categories at CT this year, and two of the examples were relations, and profunctors. Perhaps this is the sort of the universal property you're looking for?

Not really, because in her talk she talks about the 'natural' kind of closed structure one would put on a double category, that is, ask for exponentials wrt the tight direction. Here instead I'm looking for a tensor-hom adjunction in the loose direction, that is I have

$$\{A \overset{l}\to [B, C]\} \cong \{A \otimes B \overset{l}\to C\}$$

Matteo Capucci (he/him) (Aug 18 2023 at 14:54):

Mike Shulman said:

I would say it's just a bicategorical exponential in the bicategory of loose arrows.

That's ok but a bit disappointing if I know that such a bicategory is part of a double category.

So for example, the universal property I'd expect would be something like
image.png

Matteo Capucci (he/him) (Aug 18 2023 at 14:55):

where I mean the squares on the left are in natural bijection with the squares on the right

Matteo Capucci (he/him) (Aug 18 2023 at 16:00):

Making this more precise: $[-,=] : \mathbb D^{lop} \times \mathbb D \to \mathbb D$ should be a (lax?) double functor such that $- \otimes B \dashv [B, -]$. Such an adjunction, however, would imply that $[A,B]$ is an exponential in the tight direction! Something is amiss...

Nathanael Arkor (Aug 18 2023 at 16:10):

Matteo Capucci (he/him) said:

Nathanael Arkor said:

Susan Niefield gave a talk about cartesian closed double categories at CT this year, and two of the examples were relations, and profunctors. Perhaps this is the sort of the universal property you're looking for?

Not really, because in her talk she talks about the 'natural' kind of closed structure one would put on a double category, that is, ask for exponentials wrt the tight direction. Here instead I'm looking for a tensor-hom adjunction in the loose direction, that is I have

$$\{A \overset{l}\to [B, C]\} \cong \{A \otimes B \overset{l}\to C\}$$

Susan's universal property splits into a universal property for the tight morphisms and a universal property for the loose morphisms. I wouldn't be surprised if, under the assumption of fibrancy on the double category, the universal property implied the more conventional bicategorical universal property (similar to how a monoidal double category comprises monoidal structure on both the tight morphisms, and the loose morphisms, but under fibrancy implies that the loose bicategory is monoidal).

Mike Shulman (Aug 18 2023 at 16:10):

I don't know if there is a more double-categorical version of loose monoidal closedness. But there is a more double-categorical version of loose compact closedness! Unfortunately I don't think it has been written down in the literature anywhere, but @Christian Williams and I had some ideas about a way to formulate it, at least when the double category is fibrant and hence equivalent to a proarrow equipment. In the compact closed case, instead of a two-variable internal-hom functor you have a one-variable "dual" functor $\mathbb{D}^{lop} \to \mathbb{D}$ which should be vertically covariant and strong monoidal, and then there should be bijections of squales like the one you drew but with $B^o \otimes C$ on the right. But it's not entirely clear what coherence axioms should be assumed of such bijections.

Mike Shulman (Aug 18 2023 at 16:13):

However, if you change from a fibrant double category to a proarrow equipment $(-)_\bullet : \mathcal{K} \to\mathrm{Map}(\mathcal{M})$ (where Map denotes the sub-bicategory of left adjoints), then you can say that the structure of $(-)^o$ should be a [[duality involution]] on $\mathcal{K}$, a compact-closed structure on $\mathcal{M}$ which therefore induces a duality involution on $\mathrm{Map}(\mathcal{M})$, and that $(-)_\bullet$ is a morphism of duality involutions.

Matteo Capucci (he/him) (Aug 18 2023 at 18:23):

Nathanael Arkor said:

Susan's universal property splits into a universal property for the tight morphisms and a universal property for the loose morphisms. I wouldn't be surprised if, under the assumption of fibrancy on the double category, the universal property implied the more conventional bicategorical universal property (similar to how a monoidal double category comprises monoidal structure on both the tight morphisms, and the loose morphisms, but under fibrancy implies that the loose bicategory is monoidal).

I can assume fibrancy for the examples I'm interested, however this seems unlikely unless I'm missing something. For instance, Rel is a cartesian closed double category with $[A,B] = B^A$ but the loose closed structure is, on objects at least, given by products.