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Stream: learning: questions

Topic: All about 2-rigs

fosco (Jan 19 2021 at 21:05):

Lately I've been interested in studying the theory of 2-rigs deeper.

Since the terminology isn't exactly well-established, by "2-rig" I mean a category $C$ having two monoidal structures, the "additive" one $\oplus$ and the "multiplicative" one $\otimes$, and $\otimes$ distributes over $\oplus$; shortly, a 2-rig for me is what the nLab calls a rig category. A 2-rig for the nLab is another thing, alas.

the definition goes back to Laplaza, if I recall correctly, and it was recently proved by Elgueta a conjecture posed by @John Baez that says that the 2-rig of finite sets, bijections, coproducts and products is initial in a suitable 2-category of 2-rigs.

Coherence conditions for a diagram in a 2-rig are quite a mess but fortunately I can limit myself to additively cocartesian monoidal structures, i.e. where $\oplus$ is given by coproduct.

This is very interesting stuff, and I have eagerly used it in a project I've been thinking about for quite some time.

I am planning to post more material on this later (quite likely, way more questions to come: I have been trapped in a foxhole for quite some time, and I have no clue how to prove a bunch of conjectures): for the moment, my question is just one:

Many categories of usual practice are 2-rigs, so it is natural to ask, has anyone studied the notion of a category enriched over a 2-rig? If yes, the enrichment was understood with respect to the additive structure, or the multiplicative structure, alone, or in synergy with the other? How does distributivity of $\otimes$ over $\oplus$ enhances the theory of enriched categories?

I've been thinking about this with @Fabrizio Genovese @Matt Earnshaw @Daniele Palombi and a bunch of other people, hence the tags.

Nathanael Arkor (Jan 19 2021 at 23:53):

I believe @Joshua Meyers showed that if you enrich over a cocartesian monoidal structure, the hom-objects must all be isomorphic (#theory: category theory > Strange Structure), so you probably don't want to pick the additive structure.

Joe Moeller (Jan 20 2021 at 00:01):

I think if the additive structure isn't cocartesian, ie if you have two monoidal structures with distribution, then you call it a "rig category" following either Kelly or Laplaza, iirc.

Nathanael Arkor (Jan 20 2021 at 00:04):

(@fosco mentioned this in the second paragraph.)

Joe Moeller (Jan 20 2021 at 00:06):

Wow, I read the whole thing, but my eyes completely missed that. I guess I was skimming the beginning more than really reading.

Joe Moeller (Jan 20 2021 at 00:10):

I was thinking about this question also recently, specifically with Lawvere metric spaces. They are categories enriched over the tropical semiring, but this definition only talks about the multiplicative structure.

Joe Moeller (Jan 20 2021 at 00:12):

The additive structure is surely also important in certain contexts. I was brought to think about this after hearing a talk by Callum Reader last year at TICT, and then shortly after again due to a talk by Simon Willerton.

Joe Moeller (Jan 20 2021 at 00:14):

Thinking about optimization for instance will surely require you to think about both addition and min. This was the context I think in both of those talks.

Joe Moeller (Jan 20 2021 at 00:15):

So the natural question is what role does the additive structure have in a category enriched over the multiplicative structure?

fosco (Jan 20 2021 at 08:09):

in the tropical semiring $(\mathbb R, \min,+)$ the additive structure is "min", so it's coproduct in $(\mathbb R, \ge)$ (=product in $(\mathbb R, \le)$, right?

fosco (Jan 20 2021 at 08:10):

Another example that motivates my interest is enrichment over lattices, regarded as 2-rigs with respect to $\land,\lor$

fosco (Jan 20 2021 at 08:11):

but you see, in both these examples sum is idempotent, I would like to find an example that is not!

Anyway, this question acquires more sense when it's put in a broader perspective, I'll be back with it and more questions :grinning:

fosco (Jan 23 2021 at 12:25):

Enjoy my confusion and handwaving! https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Differential.20rig.20categories

fosco (Jan 23 2021 at 12:25):

(how does one link a thread "internally"?)