Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: event: Online CT seminar

Topic: July 24, 1400 GMT - Fosco Loregian - 2-(commutative algebra)

Eric M Downes (Jul 18 2024 at 16:02):

Abstract

A "2-rig" arises as a categorification of the notion of a semiring. One possible definition for it is that it is a monoidal category where each A ⊗ _ functor preserves coproducts, or more general colimits. Such a definition plays well with the goal of categorifying commutative algebra: basic theorems in this area, such as existence and properties of polynomial rings, can be explored within this framework.
The theory becomes particularly intriguing when one categorifies differential rings and studies their properties. This approach illuminates certain properties of the category of combinatorial species, which is the analogue of polynomial/power series rings in the 2-category of 2-rigs. 

I will survey some of these topics, which form an ongoing project with Todd Trimble.

Jean-Baptiste Vienney (Jul 24 2024 at 13:46):

Starting in 15 minutes folks.

fosco (Jul 24 2024 at 13:47):

I'm around here. Do I have to connect a bit in advance?

Jean-Baptiste Vienney (Jul 24 2024 at 13:48):

You can connect a bit in advance just to be sure that everything works fine i.e. that we can hear you and see your slides.

fosco (Jul 24 2024 at 15:43):

https://arxiv.org/pdf/1109.2688 this is the paper I mentioned at the end!

Jean-Baptiste Vienney (Jul 24 2024 at 15:49):

Just for completeness. JS was probably thinking about this paper:
Differential Algebras in Codifferential Categories.

Jean-Baptiste Vienney (Jul 25 2024 at 03:25):

This paper talks about differential linear logic and analytic functors which are related to combinatorial species. Still far from your notion but at least the two works can maybe meet on this example.

Monoidal bicategories, differential linear logic, and analytic functors (Fiore, Gambino, Hyland)

Jean-Baptiste Vienney (Jul 25 2024 at 04:20):

Concerning your idea of integral domain etc… maybe you could define (by the way, notice that for a lot of the notions below which already work for all monoids and not only semirings, we only need a monoidal category). Let’s say we work with a symmetric bimonoidal category. I would like to define this:

• $R$ is an integral domain: if $R$ is not reduced to $I$ and $X \otimes Y \simeq X \otimes Z \Rightarrow Y \simeq Z$.
• Unit: $X$ such that there exists $Y$ with $X\otimes Y \simeq I$.
• Associates elements: $X,Y$ are associates if there exists a unit $U$ such that $X \simeq U \otimes X$.
• $R$ is a UFD: if $R$ is an integral domain, every element $X \in R$ admits a decomposition $X=A_1 \otimes … \otimes A_n$, unique up to permutation and associates (Edit: argh, sorry the $A_n$ must be irreducibles, we must define this first).
• Ideal of $R$: subcategory $\mathcal{I}$ which is stable by $\oplus$, contains $0$ and such that if $X \in R$ and $Y \in \mathcal{I}$ then $X \otimes Y \in \mathcal{I}$.
• $R$ is Noetherian: if every chain $\mathcal{I}_1 \subseteq … \subseteq \mathcal{I}_n \subseteq …$ terminates.

It would be interesting to see if we recover the usual notions when restricting ourselves to a thin symmetric bimonoidal category which is just an ordered commutative semiring. I think it means that morphisms should be interpreted as a kind of divisibility.

Jean-Baptiste Vienney (Jul 25 2024 at 05:03):

An irreducible object should be an object $X$ such that if we have a morphism $Y \rightarrow X$ then $Y \simeq X$ or $Y \simeq I$. It makes me think to the notion of irreducible representation.

Jean-Baptiste Vienney (Jul 25 2024 at 05:17):

And prime element makes me think to indecomposable representation.

Zoltan A. Kocsis (Z.A.K.) (Jul 25 2024 at 06:07):

Jean-Baptiste Vienney said:

An irreducible object should be an object $X$ such that if we have a morphism $Y \rightarrow X$ then $Y \simeq X$ or $Y \simeq I$. It makes me think to the notion of irreducible representation.

Very tangential fun fact, but I thought I'd mention it while we're in "reminds me of" mode: each finite Heyting algebra has exactly one element $r$ which satisfies the implicative irreducibility condition: $x\rightarrow y=r$ implies $x=\top,y=r$ i.e. $r$ can be expressed as an implication in exactly one way.

The properties of this element determine those of the algebra to some extent: e.g. the implicatively irreducible element is double negation stable precisely in Boolean algebras (and then it's also central). IIRC in infinite HAs the implicatively irreducible element is still unique, but may fail to exist.

fosco (Jul 25 2024 at 09:50):

Thanks for the comments! What I've been thinking about ideals is that it would be nice to have a correspondence between them (it's easy to define an ideal as a subcategory such that... as @Jean-Baptiste Vienney did above) and quotients, meaning coinverters along a family of initial maps $\varnothing \Rightarrow P\otimes-$ for some elements P generating the ideal.

fosco (Jul 25 2024 at 09:52):

This opens many questions: every kernel of a 2-rig homomorphism is an ideal; is the converse true? Is this some categorified notion of regular epimorphism?