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Stream: learning: reading & references

Topic: Cartesian enriched categories

Vincent Moreau (Sep 10 2024 at 10:49):

Dear category theorists,

Let $V$ be a monoidal cartesian category (not necessarily closed or complete). I am looking for a simple description of a notion of "$V$-enriched terminal object" and "$V$-enriched binary cartesian product", if such things exist. Following this answer, if $C$ is a $V$-enriched category, one can use the cartesianness of the enriching base $V$ to define $V$-functors

$$C \ \longrightarrow\ 1 \qquad\text{and}\qquad C \ \longrightarrow\ C \times C$$

which makes it possible to ask for these functors to have $V$-right adjoints. This specializes to the usual notions when $V = \mathbf{Set}$, but I have not found this written down in a paper.

I am aware that the interesting concept of weighted limit is central in enriched category theory, and that it provides the right notion of limit for the enriched setting, but I am looking for references which spell out in accessible terms what are enriched finite cartesian products. I have the impression that this should be a very simple example -- and enriched cartesian products are even mentioned on the nlab page for enriched CCCs -- yet I can't find such a reference.

Do anyone know a reference that I could cite? or is this notion more complicated than I thought, and perhaps that there are multiple, non-equivalent generalizations to the enriched setting? or something else?

fosco (Sep 10 2024 at 10:55):

FWIW, I would follow a similar path and define a "object with Cartesian product" in a $(V,\times,1)$-category as a $C$ such that $C\to 1$ and $C\to C\times C$ both have a right adjoint.

Vincent Moreau (Sep 10 2024 at 12:19):

fosco said:

FWIW, I would follow a similar path and define a "object with Cartesian product" in a $(V,\times,1)$-category as a $C$ such that $C\to 1$ and $C\to C\times C$ both have a right adjoint.

Did you meant in $V\text{-}\mathbf{Cat}$? Then this coincides with the notion of cartesian object, and as stated in my message I think this should give a good notion of cartesian $V$-category, when $V$ is cartesian monoidal.

What puzzles me is that I cannot find a place in the literature which introduces cartesian $V$-categories, in this way or another, and not even the notion of enriched terminal object.

Nathanael Arkor (Sep 10 2024 at 13:04):

Street defines the notion of cartesian V-category in section 3 of Kan extensions and cartesian monoidal categories even when V is not cartesian. However, it's not immediately clear to me that there necessarily exists a class of weights $\Phi$ for which a V-category is $\Phi$-complete if and only if it is cartesian in this sense.

Mike Shulman (Sep 10 2024 at 13:50):

Isn't this just the case of "conical" $V$-limits (section 3.8 in Kelly's book) when the domain ordinary category is empty or is $2$?

Vincent Moreau (Sep 10 2024 at 15:10):

Nathanael Arkor said:

Street defines the notion of cartesian V-category in section 3 of Kan extensions and cartesian monoidal categories even when V is not cartesian. However, it's not immediately clear to me that there necessarily exists a class of weights $\Phi$ for which a V-category is $\Phi$-complete if and only if it is cartesian in this sense.

I like the generality of this approach, but I have a naive question: in the case of plain categories, being cartesian is a property, as this boils down to the diagonal functor having a right adjoint, which is unique up to iso if it exists. In the enriched case described in the article you have linked, we only ask for the given $V$-enriched functor $C \otimes C \to C$ to have a left adjoint, whatever it is. Therefore, I am not sure if two such "cartesian monoidal" structures on the same $V$-enriched category must always be the same, for the appropriate notion of sameness, given that we don't have a common left adjoint to compare them.

Vincent Moreau (Sep 10 2024 at 15:25):

Mike Shulman said:

Isn't this just the case of "conical" $V$-limits (section 3.8 in Kelly's book) when the domain ordinary category is empty or is $2$?

Perhaps this coincides with the description given in terms of enriched right adjoints to the functor into the terminal enriched category and the diagonal functor, I don't know. I'm trying to see if there is a reference which defines such a notion of cartesian enriched category for a cartesian monoidal base, which I could cite. Perhaps this is folklore?

Nathanael Arkor (Sep 10 2024 at 15:29):

Vincent Moreau said:

Therefore, I am not sure if two such "cartesian monoidal" structures on the same $V$-enriched category must always be the same, for the appropriate notion of sameness, given that we don't have a common left adjoint to compare them.

That's true; cartesianness is a property of a specific monoidal structure in this setting.

Vincent Moreau (Sep 10 2024 at 15:31):

This is interesting, how do you see that?

Nathanael Arkor (Sep 10 2024 at 15:32):

Sorry, my phrasing was unclear. I meant that you're right that being "cartesian" is a property of a given monoidal category, rather than a given category. So prima facie there may be two different monoidal structures on a V-category, both of which are cartesian, but which are not related.

Vincent Moreau (Sep 10 2024 at 15:33):

Ah sorry, I see!

Nathanael Arkor (Sep 10 2024 at 15:34):

Vincent Moreau said:

Perhaps this coincides with the description given in terms of enriched right adjoints to the functor into the terminal enriched category and the diagonal functor, I don't know.

I think it does coincide, at least under the assumption free V-categories on ordinary categories exists. Without that assumption, one can still use Kelly's explicit definition of "conical limit" (which gives the universal property you're looking for), but it's no longer necessarily a special kind of weighted limit, as far as I can see.

Vincent Moreau (Sep 10 2024 at 15:39):

OK, well I cannot really afford to have a cocomplete enriching base so I'm going to dive into section 3.8 and try to work it out. Thanks!

Nathanael Arkor (Sep 10 2024 at 15:42):

For what it's worth, the definition as a cartesian object in V-Cat appears as Definition 3.9 of Modality via Iterated Enrichment.

Vincent Moreau (Sep 10 2024 at 15:49):

Indeed, thanks!

Mike Shulman (Sep 10 2024 at 17:42):

If $V$ isn't cocomplete then I think you can just work with $\widehat{V}$-categories and their weighted limits.

Vincent Moreau (Sep 11 2024 at 15:34):

Nathanael Arkor said:

Sorry, my phrasing was unclear. I meant that you're right that being "cartesian" is a property of a given monoidal category, rather than a given category. So prima facie there may be two different monoidal structures on a V-category, both of which are cartesian, but which are not related.

I've thought a bit about that. Suppose for the moment that we work in $V = \mathbf{Set}$, and that $(C, \otimes, I)$ is a locally small monoidal category which is cartesian in the sense of this article of Street. This means that $I$ is terminal and that we have an adjunction

$$? \quad:\quad C \ \rightleftarrows\ C^2 \quad:\quad \otimes$$

with $\otimes$ the right adjoint. A priori, we don't know anything on the functor $?$, and especially not that it is the diagonal functor. But now, we compose this with the adjunction

$$\pi_1 \quad:\quad C^2 \ \rightleftarrows\ C \quad:\quad (-, I)$$

where the right adjoint is the functor $c \mapsto (c, I)$.

But now, thanks to the fact that $I$ is the monoidal unit, the composite of the two right adjoints is isomorphic to the identity endofunctor of $C$, whereas the composite of the two left adjoints is the functor

$$c \mapsto \pi_1(?(c)) \quad:\quad C \ \longrightarrow\ C^2 \ \longrightarrow\ C$$

which is then isomorphic to the identity endofunctor. By doing this story again with $\pi_2$, we get that the mysterious left adjoint $?$ is actually the diagonal.

Vincent Moreau (Sep 11 2024 at 15:35):

If all of this is correct, then this means that this notion of monoidal cartesian enriched category used by Street boils down to the usual notion when we enrich in sets, which is a nice thing!

Vincent Moreau (Sep 11 2024 at 15:37):

I'm not fluent enough in enriched categories to assert that this extends to the enriched case, but I wouldn't be shocked if it does in the case where the enriching base $V$ is cartesian, which makes it possible to have a terminal $V$-category and hence to talk about the $\pi_i$.

Nathanael Arkor (Sep 11 2024 at 16:28):

Vincent Moreau said:

If all of this is correct, then this means that this notion of monoidal cartesian enriched category used by Street boils down to the usual notion when we enrich in sets, which is a nice thing!

Street explains why it recovers the usual definition for V = Set, but I don't see that Street's definition should coincide with a cartesian object in V-Cat when V is not cartesian monoidal (meaning the tensor product is the cartesian product, not simply that V has finite products).

Vincent Moreau (Sep 11 2024 at 17:02):

Sadly I don't understand where Street explains this

Vincent Moreau (Sep 11 2024 at 17:03):

In any case, I have some hope for the proof above to hold in the case of $V$-enriched categories for $V$ cartesian monoidal

Kevin Carlson (Sep 11 2024 at 17:03):

Yes, I think it goes through fine in that case.

Nathanael Arkor (Sep 11 2024 at 17:14):

Vincent Moreau said:

Sadly I don't understand where Street explains this

image.png
Here, he shows that the tensor product of the monoidal V-category in question satisfies this universal property, and when V is cartesian monoidal, this is the universal property of a product.

Nathanael Arkor (Sep 11 2024 at 17:18):

But I think your argument is also clear.

Vincent Moreau (Sep 11 2024 at 18:39):

Nathanael Arkor said:

image.png
Here, he shows that the tensor product of the monoidal V-category in question satisfies this universal property, and when V is cartesian monoidal, this is the universal property of a product.

I understand that $\mathscr{A}$ is endowed with a comonoidal structure, which when passed to the monoidal functor ob from $V$-Cat to Set yields a comonoid, hence the fact that $\Delta$ is the diagonal on objects. I understand too that the naturality of the isomorphisms in your screenshot is formulated with $\Delta$ in its $A$ variable. However, does this show that $\Delta$ is isomorphic to the diagonal functor in the case where $V$ is cartesian monoidal? This I don't see.

I suppose that this is what the conical limit defining products that you and Mike were referring to boils down to, by the way?

Nathanael Arkor (Sep 11 2024 at 18:50):

When V is cartesian monoidal, the tensor product of V-categories is also cartesian, and so every V-category has a unique comonoid structure with respect to the tensor product. Thus $\Delta$ is not just the diagonal on objects, but also on morphisms.

Vincent Moreau (Sep 11 2024 at 19:10):

Right, of course! Thanks for the explanation, Nathanael :)

Mike Shulman (Sep 11 2024 at 21:25):

I feel like the name "cartesian monoidal enriched category" for this thing is kind of misleading when $V$ is noncartesian. I would expect a "cartesian monoidal category" to always be one where the monoidal product is the cartesian product, i.e. adjoint to the standard diagonal $A\to A\times A$, not adjoint to some other "diagonal" $A\to A\otimes A$ that's given as extra data. I would be inclined to give this a name with something like "Hopf" in it.

Nathanael Arkor (Sep 11 2024 at 22:14):

It's a "comap pseudomonoid" following the terminology of Day and Street.

Mike Shulman (Sep 12 2024 at 00:22):

I suppose.