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Stream: learning: questions

Topic: quasi-cartesian monoidal categories

Morgan Rogers (he/him) (Mar 10 2021 at 10:03):

In a cartesian monoidal category, the (monoidal) product comes equipped with the canonical projection maps,
$X \leftarrow X \otimes Y \rightarrow Y,$
(and dually for a cocartesian monoidal category). Is there a known axiomatisation of monoidal categories equipped with such maps, perhaps extracting some of the identities satisfied in the cartesian case?

Matteo Capucci (he/him) (Mar 10 2021 at 10:05):

I know this article goes into the question https://golem.ph.utexas.edu/category/2016/08/monoidal_categories_with_proje.html

Matteo Capucci (he/him) (Mar 10 2021 at 10:06):

Apparently, you get a semicartesian monoidal cat

Morgan Rogers (he/him) (Mar 10 2021 at 10:07):

That's exactly what I wanted! Thank you!

Fawzi Hreiki (Mar 10 2021 at 10:43):

A fun exercise is proving that projections and diagonals together imply that the tensor is the cartesian product.

Morgan Rogers (he/him) (Mar 10 2021 at 10:46):

I notice that Mike Shulman made a similar comment under that ncafe post. Can you be more specific? Which things is the diagonal required to commute with? I don't immediately see how you could get a universal property out.

Morgan Rogers (he/him) (Mar 10 2021 at 10:49):

For example, I can imagine a monoidal category with projections where the monoidal product is idempotent (so we can use the morphism $X \cong X \otimes X$ as a diagonal) and it doesn't seem like that will typically be cartesian, if non-trivial examples exist.

Robin Piedeleu (Mar 10 2021 at 10:57):

A monoidal category is cartesian iff it is equipped with a sufficiently nice comonoid structure on each object, whose comultiplications and counits are natural. The sufficiently nice condition roughly says that the comultiplication (resp. counit) over $X\otimes Y$ is the monoidal product of the comultiplications (resp.counits) over $X$ and $Y$. This is best proved using string diagrams! In fact it is sufficient for the counit to be natural (without the naturality of the comultiplication but the rest is the same) for this equivalence to hold if I remember correctly.

Fawzi Hreiki (Mar 10 2021 at 11:02):

Yeah, just the diagonal and projection maps aren't enough of course to get the UMP. You need naturality conditions on them.

Robin Piedeleu (Mar 10 2021 at 11:04):

Actually, I'm wondering now if the sufficiently nice condition + naturality boil down to the comultiplication (aka diagonal) and counit (aka projections) being monoidal natural transformations $Id \rightarrow Id \otimes Id$ and $Id \rightarrow I$ (with $I$ the unit of the monoidal product).

Fawzi Hreiki (Mar 10 2021 at 11:08):

That's the definition, I believe.

Robin Piedeleu (Mar 10 2021 at 11:17):

The details can be found in Chapter 6 of these notes: https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf where the authors make my vague "sufficiently nice" condition precise (they call it uniform copying and deleting).

Morgan Rogers (he/him) (Mar 10 2021 at 11:19):

Yeah it turns out that if the monoidal product is idempotent, it forces the category to be posetal, and the product to be the meet, so it is indeed cartesian!

Fawzi Hreiki (Mar 10 2021 at 11:24):

A nice fact is that an object in a category is product idempotent iff it is subterminal, so the product idempotents are 'truth values'

Fawzi Hreiki (Mar 10 2021 at 11:24):

But I'm sure you knew that.

Morgan Rogers (he/him) (Mar 10 2021 at 11:36):

Right, but I was expecting it to be possible to have an idempotent non-cartesian monoidal product where the objects don't have to be subterminal. Turns out that with projections that's not possible!

Nathanael Arkor (Mar 10 2021 at 13:11):

Morgan Rogers (he/him) said:

Yeah it turns out that if the monoidal product is idempotent, it forces the category to be posetal, and the product to be the meet, so it is indeed cartesian!

I may be missing the obvious, but what makes this true?

Morgan Rogers (he/him) (Mar 10 2021 at 13:41):

I realise now that I was employing the extra assumption that the projection maps decompose tensored morphisms, but now I'm not even sure that assumption makes sense :scream:

Morgan Rogers (he/him) (Mar 10 2021 at 13:49):

Okay, so if we have idempotency and $X \cong X \otimes X \to X \otimes 1 \cong X$ is the identity for all $X$, and similarly for the other projection, the posetal part of what I said is true, because given $f,g:X \rightrightarrows Y$ we can tensor them together and get $f \otimes g: X \otimes X \to Y \otimes Y$ corresponding to some $t:X \to Y$ via the diagonal isomorphism, and then $f = t = g$ by going along the projection morphisms.

Morgan Rogers (he/him) (Mar 10 2021 at 13:51):

And then if $P$ admits morphisms $a:P \to X$ and $b: P \to Y$ then we have $P \cong P \otimes P \xrightarrow{a\otimes b} X \otimes Y$ being the (necessarily unique) product factorisation.

Morgan Rogers (he/him) (Mar 10 2021 at 13:56):

Morgan Rogers (he/him) said:

If we have idempotency and $X \cong X \otimes X \to X \otimes 1 \cong X$ is the identity for all $X$...

The factoring map argument works without the second assumption, but it's not obvious to me that it should be unique without it.
Unfortunately for me, the situation I'm dealing with does have this property, and so I realised that the tensor product is only functorial on a posetal subcategory of the category I actually care about, which somewhat dampened my interest in these things.

Nathanael Arkor (Mar 10 2021 at 13:56):

A possibly interesting corollary is that the "idempotent coproduct cocompletion" forms a KZ doctrine that is not fully faithful, which is relatively unusual.

Nathanael Arkor (Mar 10 2021 at 13:58):

(Though the situation simplifies there because you actually have that the tensor product is the coproduct.)

John Baez (Mar 10 2021 at 15:36):

I usually think of a semicartesian monoidal category as one where the unit object is terminal. I'd forgotten that Tom Leinster discussed this alternative (equivalent, but less elegant) definition in terms of projections. Thanks for pointing it out!

John Baez (Mar 10 2021 at 15:41):

My favorite semicartesian categories are the category of convex sets and convex-linear maps, and the category of Poisson manifolds and Poisson maps. I like these examples because they're interesting even to mathematicians who don't particularly care about category theory: they are not manufactured, but "found in nature".

Fawzi Hreiki (Mar 10 2021 at 15:44):

Likewise, the category of affine spaces motivated the name 'affine logic' for what is effectively the internal logic of semicartesian closed monoidal categories.

John Baez (Mar 10 2021 at 15:45):

Yes, affine spaces are another one of my favorites - thanks for reminding me, I haven't thought about this stuff recently.

Mike Shulman (Mar 10 2021 at 16:21):

Another "found in nature" semicartesian monoidal category is the poset of nonnegative real numbers $([0,\infty],\ge)$, with addition as the monoidal structure. This is the category such that $[0,\infty]$-enriched categories are (Lawvere generalized) metric spaces.