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: learning: questions

Topic: ✔ self-enrichment vs monoidal closed

Mike Stay (Jun 08 2023 at 18:49):

A monoidal closed category is canonically self-enriched. Is it always the case that a self-enriched category is monoidal closed? If C is closed and self-enriched, we can reconstruct the monoidal structure. But if C is monoidal and self-enriched, is it always possible to turn it into a monoidal closed category where the monoidal product of the closed category may be different?

For example, we can enrich the category of pointed sets over itself by taking the point of each hom set to be the function that maps everything to the point in the target. The enrichment works with the cartesian product even though homming with an object X isn't right adjoint to taking the product with X in PSet. However, there's another monoidal product, called the smash product, that we get from the cartesian product by modding out by a relation. The smash product with X is left adjoint to homming with X. When we mod out by the relation, the composition function behaves exactly the same; similarly, the monoidal unit object for the cartesian product is the same as for the smash product. So even though PSet is monoidal under the cartesian product, can be enriched over itself using that monoidal structure, and is not closed relative to that structure, we can "massage" it into a true closed structure.

This stackoverflow question gives the example of Top, where we can self-enrich by giving the hom set the discrete topology, but products in Top don't generally have right adjoints. However, it leaves open the possibility that there is some other monoidal product that would be left adjoint to the internal hom.

If a category is self-enriched, is it always possible to "massage" the monoidal structure and internal hom structure to make them be adjoints?

Nathanael Arkor (Jun 08 2023 at 18:55):

What are you taking for your definition of "self-enriched"?

Nathanael Arkor (Jun 08 2023 at 18:58):

From the context, I presume you mean a monoidal category $\mathcal V$ together with a $\mathcal V$-category $\mathscr V$ such that $\mathscr V_0 = \mathcal V$?

Nathanael Arkor (Jun 08 2023 at 19:24):

I may be missing something, but I don't even see that a self-enriched category in this sense is closed. In particular, I don't see why we should have $\mathscr V(I, A) \cong A$ (for $I$ the unit of $\mathcal V$, and $A$ an arbitrary object of $\mathcal V$), which is required for closure.

Mike Shulman (Jun 08 2023 at 19:34):

Note also that every [[closed category]] is enriched over itself in the sense that any category can be enriched over a closed category, but it may not have any monoidal structure.

Mike Stay (Jun 08 2023 at 21:59):

Thanks, both. Let me refine my question, then.

1. a. Suppose (V, ⊗, I) is monoidal and V can be enriched in (V, ⊗, I) as described by Nathanael. Does there necessarily exist some ⊠, J such that (V, ⊠, J) is monoidal closed?
2. b. What if V is symmetric?
3. c. If no to both of the above, what more is necessary for the hom/tensor adjunction to work?
4. Suppose (V, ⊸) is closed. When does it have monoidal structure? Reconstruction of monoidal structure says that it suffices for [A, [B, -]] to be a representable V-functor. Is it necessary?

Mike Shulman (Jun 08 2023 at 23:00):

4. Yes, it's necessary, because this isomorphism holds in any closed monoidal category.

John Baez (Jun 08 2023 at 23:17):

I've always believed the answer to questions 1 is "no", i.e. not every monoidal category with an enrichment over itself is closed, because closed requires an adjunction between the internal hom and the tensor product

$\mathrm{hom}(x \otimes y, z) \cong \mathrm{hom}(x, [y,z])$

and I don't see how you'd get that starting from an arbitrary self-enriched category!

John Baez (Jun 08 2023 at 23:18):

Nathanael's counterargument may be easier.

John Baez (Jun 08 2023 at 23:19):

But either way, we should come up with some counterexamples that actually arise naturally in mathematics!

Mike Shulman (Jun 08 2023 at 23:24):

Re 1 and 2, for the example of $(\rm Top,\times,1)$ which can be self-enriched with hom-spaces $[X,Y] = {\rm Top}(X,Y)$ with the discrete topology, these hom-spaces do not have left adjoints. In particular, the functor $[1,-] : \rm Top \to Top$ is the functor that retopologizes any space with the discrete topology, and this is not a right adjoint because it doesn't preserve limits: infinite limits of discrete spaces are not in general discrete. This isn't quite a proof that there is no closed monoidal structure on $\rm Top$ -- I'm not even sure how one could go about proving that -- but it's a pretty strong indication that this won't always be the case, since if the given tensor product doesn't have a right adjoint and the given internal-hom doesn't have a left adjoint, how could you hope to produce both a new tensor product and a new internal-hom that are adjoint to each other?

Mike Shulman (Jun 08 2023 at 23:25):

Re: 3, are you after something other than the obvious

$$\mathrm{hom}(x \otimes y, z) \cong \mathrm{hom}(x, [y,z])$$

?

John Baez (Jun 08 2023 at 23:26):

Yeah, I was going to suggest that as the answer to 3. "It's true if it's... true."

:upside_down:

John Baez (Jun 08 2023 at 23:29):

I think Mike's question 1 becomes "unfair" when he allows us to pick a new tensor product $\boxtimes$ on our self-enriched monoidal category when we're trying to make it into a closed monoidal category. (Is he also allowing the internal hom in the closed monoidal category to be different from the hom in the self-enrichment? Probably not, but with a guy like this you never know.)

Mike Stay (Jun 10 2023 at 16:35):

John Baez said:

(Is he also allowing the internal hom in the closed monoidal category to be different from the hom in the self-enrichment? Probably not, but with a guy like this you never know.)

In the PSet case above, is the internal hom when enriching over Set different from the hom in the self-enrichment? On one hand, obviously: in the first, composition uses the cartesian product of sets, and in the second, composition uses the smash product of pointed sets. On the other hand, the pointed sets are the plain sets equipped with structure (the point) that the morphisms for composition and identity happen to preserve.

Notification Bot (Jun 12 2023 at 19:04):

Mike Stay has marked this topic as resolved.