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Stream: learning: questions

Topic: linearly distributive functors between *-autonomous cats

Matteo Capucci (he/him) (Oct 25 2025 at 08:32):

Does anyone know if the sub-2-category of symmetric [[linearly distributive categories]] spanned by [[star-autonomous]] categories is equivalent to that of symmetric *-autonomous categories with lax monoidal functors which also laxly preserve the duality, i.e. those equipped with a coherent map $F(c^*) \to F(c)^*$?

Max Demirdilek (Jan 25 2026 at 12:50):

@Matteo Capucci: Apologies for taking a while to share my promised comment :upside_down:

In Theorem 2.94 of my recent preprint, I show that the $(2,1)$-category of braided linearly distributive (LD) categories with negation, braided Frobenius linearly distributive (LD) functors, and morphisms between them is $2$-equivalent to the $(2,1)$-category of braided Grothendieck-Verdier (GV) categories, braided Grothendieck-Verdier (GV) functors, and morphisms between them (apologies for the self-citation). Recall that (non-symmetric) star-autonomous categories are also known as GV-categories. 

This $2$-equivalence restricts to the symmetric setting: the $(2,1)$-categories of symmetric LD-categories with negation and symmetric GV-categories are $2$-equivalent. As expected, these are those full sub-$(2,1)$-categories given by braided LD-categories with negation (respectively, GV-categories) whose two braidings (respectively, single braiding) are symmetric.

Informally, a (braided) GV-functor between (braided) GV-categories is a (braided) lax monoidal functor equipped with a family of isomorphisms that witness the functor’s strong preservation of left and right duals. See Definition 2.39 for the precise formulation.  

The notion of functor suggested in your post appears to be more general. You ask for

lax monoidal functors which also laxly preserve the duality.

Any lax monoidal functor $F\colon \mathcal{C}\to \mathcal{D}$ between GV-categories $(\mathcal{C},K)$ and $(\mathcal{D},k)$, together with a morphism $F(K)\to k$ (what I call a form), induces morphisms $F(X^{\ast})\to F(X)^{\ast}$ and $F({^{\ast}X})\to {^{\ast}F(X)}$ for all $X\in \mathcal{C}$ (Definition 2.35). It seems that the resulting $2$-category of (symmetric) GV-categories, (symmetric) lax monoidal functors equipped with a form, and morphisms between them (Definition 2.40) is $2$-equivalent to the $2$-category of (symmetric) LD-categories with negation, (symmetric) $\otimes$-lax monoidal functors equipped with a form/counit morphism, and morphisms between them (again Definition 2.40). I haven’t checked the last claim carefully, though.

Nathanael Arkor (Jan 25 2026 at 14:59):

This is a little off-topic, but can I ask why you call them "Grothendieck–Verdier categories", when they were not introduced by Grothendieck or Verdier, and they been known as *-autonomous categories for almost 50 years?

Nathanael Arkor (Jan 25 2026 at 15:01):

(I think it is entirely perplexing that Boyarchenko and Drinfeld decided to introduce a new name for the concept and one that, it could be argued, is disrespectful towards the person who introduced and developed much of the theory of *-autonomous categories.)

John Baez (Jan 25 2026 at 19:02):

I find the name "Grothendieck-Verdier category" quite annoying; I've met Drinfeld a few times and he seems like a great guy, but he's also known for being very bad at choosing names of things (like quasitriangular Hopf algebras and arguably shtukas, though "shtuka", which is Russian for "thingie", is arguably so bad it's good).

In the paper that introduced the term Grothendieck-Verdier category, he and Boyarchenko do mention that they're the same as star-autonomous categories, so they can't even plead ignorance as an excuse. I can't shake the feeling that they're saying "forget what those category theorists did, let's name this concept after some famous algebraic geometers".

Matteo Capucci (he/him) (Jan 26 2026 at 22:14):

Thanks @Max Demirdilek for your remarks!

Matteo Capucci (he/him) (Jan 26 2026 at 22:15):

Max Demirdilek said:

Any lax monoidal functor $F\colon \mathcal{C}\to \mathcal{D}$ between GV-categories $(\mathcal{C},K)$ and $(\mathcal{D},k)$, together with a morphism $F(K)\to k$ (what I call a form), induces morphisms $F(X^{\ast})\to F(X)^{\ast}$ and $F({^{\ast}X})\to {^{\ast}F(X)}$ for all $X\in \mathcal{C}$ (Definition 2.35). It seems that the resulting $2$-category of (symmetric) GV-categories, (symmetric) lax monoidal functors equipped with a form, and morphisms between them (Definition 2.40) is $2$-equivalent to the $2$-category of (symmetric) LD-categories with negation, (symmetric) $\otimes$-lax monoidal functors equipped with a form/counit morphism, and morphisms between them (again Definition 2.40). I haven’t checked the last claim carefully, though.

This is also what I ended up concluding

Max Demirdilek (Feb 02 2026 at 08:27):

That’s a fair question, and I understand why the naming can be irritating.

My choice of terminology is not intended to be disrespectful in any way. To my knowledge, the notion of a ribbon structure on a Grothendieck-Verdier category was introduced by Boyarchenko and Drinfeld. It has since played an important role in developments in representation theory and quantum topology. In that setting, the term ribbon Grothendieck-Verdier category (and similarly braided or pivotal) is widely used. Replacing it with "ribbon (non-symmetric) *-autonomous category" would make communication with that audience rather difficult.

I don’t read the terminology as a claim about priority or as an attempt to overwrite earlier work. My intention here is simply to follow the usage most familiar in the context in which I am working.