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Stream: learning: questions

Topic: nerve of a monoidal category with deletes

Tim Hosgood (Aug 28 2021 at 19:48):

(let me prefix this question with this statement: I'm not even sure this question is well formed)

say we have a (symmetric, maybe) monoidal category which has some notion of deletion (thinking diagrammatically here). is there a notion of nerve where the $n$-simplices are $n$-fold tensor products, and the maps that lower dimension are given by deleting an element (and the maps that raise dimension given by either duplicating an element or inserting the unit element — whichever works)?

Tim Hosgood (Aug 28 2021 at 19:49):

the usual notion of nerve for a symmetric monoidal category isn't quite this: for example, the $2$-simplices look like morphisms $x_{02}\to x_{01}\otimes x_{12}$

Morgan Rogers (he/him) (Aug 29 2021 at 15:42):

Have you tried it? Are you looking for a reference or just some insight into whether it could work in principle?

Tim Hosgood (Aug 29 2021 at 22:18):

i haven't really sat down and tried it yet, because i was expecting it to be something that was already considered somewhere, so i was hoping for a reference ideally

Tim Hosgood (Aug 29 2021 at 22:18):

but if nobody knows of one, then maybe a small comment on whether or not this seems like a "good" definition would be appreciated :-)

Alexander Campbell (Aug 30 2021 at 00:41):

@Tim Hosgood I don't think I understand the question. In particular, I don't know what "deletion" means.

John Baez (Aug 30 2021 at 02:32):

"Deletion" refers to a natural transformation $\epsilon_x : x \to I$ from an object $x$ to the unit object $I$ in a monoidal category.

John Baez (Aug 30 2021 at 02:33):

Most famously, in a cartesian category $I$ is terminal so we can take $\epsilon_x$ to be the unique morphism from $x$ to $I$.

John Baez (Aug 30 2021 at 02:34):

There's a characterization of cartesian categories in terms of duplication (the diagonal) and deletion.

Amar Hadzihasanovic (Aug 30 2021 at 05:41):

If $M$ is a monoidal category, every comonoid in $M$ gives a simplicial object $\Delta^\mathrm{op} \to M$ (because comonoids are exactly monoidal functors from the augmented simplex category with the join as monoidal product).

Supposing every object $b$ in $M$ comes equipped with a comonoid, which you can see as a “copy-deletion” pair, you can associate to it this simplicial object $\Delta^\mathrm{op} \to M$; composing with the Yoneda embedding you get a functor $\Delta^\mathrm{op} \to [M^\mathrm{op}, \mathrm{Set}]$, which corresponds by currying to a functor $M^\mathrm{op} \to [\Delta^\mathrm{op}, \mathrm{Set}]$, i.e. a parametrised family of simplicial sets.
This sends each object $a$ of $M$ to the simplicial set whose $n$-simplices are elements of $\mathrm{Hom}(a, b^{\otimes n})$, i.e. “generalised elements” of the $n$-fold tensor product of $b$, and the dimension-lowering maps are what you get by using the counit (“deletion”).

If you want to just have a single simplicial set, you have a few options (taking only “global elements”, taking a colimit...)

If you just have the “deletion” and not the “copy”, you only get a semisimplicial set in this way (only faces, no degeneracies). For the degeneracies to work with the faces you need a comonoid structure.

Tobias Fritz (Aug 30 2021 at 07:13):

I don't have anything to add about the original question, but perhaps it'll help to point out that a "monoidal category with deletes" is usually called a semicartesian monoidal category. Being semicartesian is property rather than structure, namely the property of the monoidal unit being terminal! This is analogous to how being a "monoidal category with deletes and copies" is a property, namely the property of being cartesian.

Tim Hosgood (Aug 30 2021 at 09:56):

@Amar Hadzihasanovic that's very helpful, thank you!

Tim Hosgood (Aug 30 2021 at 09:57):

@Tobias Fritz I knew there had to be a name for it, and in hindsight it's the "obvious" one, but thanks :-)

John Baez (Aug 30 2021 at 16:04):

I used to call a monoidal category with duplication but not necessarily deletion a hemicartesian monoidal category, but someone at the nLab decided to call it a relevance monoidal category.

John Baez (Aug 30 2021 at 16:06):

I don't really like "relevance" since it's biased toward a specific intended application to logic, but it's certainly more descriptive than "hemicartesian".

Mike Shulman (Aug 30 2021 at 17:15):

The nLab didn't invent that terminology.

John Baez (Aug 30 2021 at 17:31):

I didn't say they did. And you didn't say I said that they did. And I didn't say that you said that I said that they did.

I guess what bugs me is that now the two obvious weakenings of the concept of "cartesian monoidal category" are called "semicartesian" and "relevance" - awkwardly asymmetrical. Oh well.

Jules Hedges (Aug 30 2021 at 18:25):

Incidentally, I normally call monoidal + deletion "affine monoidal" after affine logic, which is the other strengthening of linear logic along with relevance logic