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Jean-Baptiste Vienney (Mar 12 2026 at 14:05):I’m trying to prove a strictification theorem for differential categories as a consequence of the result that for every symmetric monoidal category there exists a symmetric monoidal equivalence between this symmetric monoidal category and a symmetric strict monoidal category. However, I don’t think this is something specific about differential categories. Let me explain.
A differential category is defined as a symmetric monoidal category enriched over commutative monoids (let’s call this first part additive symmetric monoidal category) together with a bunch of functors and natural transformations which satisfy some identities.
I’ve already proved that for every additive symmetric monoidal category there exists an additive symmetric strict monoidal category and an equivalence of additive symmetric monoidal categories between these two additive symmetric monoidal categories.
For this, it suffices to define correctly the notions of additive symmetric monoidal functors and additive symmetric monoidal natural transformations, use the strictification theorem for symmetric monoidal categories and transfer the additive structure through the symmetric monoidal equivalence.
I’m trying something similar with differential catgeories. I’ve defined differential functors, differential natural transformations and differential equivalences by adding the appropriate additional structure and axioms to the notions of additive symmetric monoidal functors, natural transformations and equivalences. But this takes a lot of time, even if it’s not very difficult because I must use quite big commutative diagrams to prove a bunch of identities.
I can give the precise definition of a differential category if needed but perhaps a strictification theorem exist for all additive symmetric monoidal categories “with a bunch of functors and natural transformations satisfying some identities”? so that the precise definition is actually not needed.
Nathanael Arkor (Mar 12 2026 at 14:07):If you can describe differential categories as the algebras for a 2-monad, then you can probably appeal to the general coherence theorem in Lack's Codescent objects and coherence.
Jean-Baptiste Vienney (Mar 12 2026 at 14:08):Awesome!
Nathanael Arkor (Mar 12 2026 at 14:08):(And being able to describe a structure in terms of a bunch of covariant functors and natural transformations is a good indication it will be describable as an algebra for a 2-monad.)
Jean-Baptiste Vienney (Mar 12 2026 at 14:10):I find it difficult to find references about 2-monads. Is there a place where it is explained how categories, symmetric monoidal categories or enriched categories for instance can be described as algebras for a 2-monad?
Matteo Capucci (he/him) (Mar 12 2026 at 14:20):A good place to start is the 2-categories companion. Then follow the references in the relevant sections for more.
Nathanael Arkor (Mar 12 2026 at 14:20):Section 5 of Lack's A 2-categories companion is one reference.
Matteo Capucci (he/him) (Mar 12 2026 at 14:20)::)
Jean-Baptiste Vienney (Mar 12 2026 at 14:20):Ok thank you!
Jean-Baptiste Vienney (Mar 12 2026 at 14:21):I’m going to look at these two references. I’ll let you know if I succeed in my goal :)
Jean-Baptiste Vienney (Mar 12 2026 at 15:02):(it's going to take me some time)
Jean-Baptiste Vienney (Mar 15 2026 at 23:05):I will give a talk at Macquarie on Wednesday and I mentioned this strictification question in my slides. I wrote « Nathanael Arkor and Matteo Capucci gave me these two references: … I’m scared by Australian 2-category theory. Please help me! » I think Steven Lack will be in the audience (he is a professor there) so we’ll see what happens :joy:
John Baez (Mar 15 2026 at 23:39):It's good to learn Lack's work on 2-categories, and luckily he is quite friendly.
Matteo Capucci (he/him) (Mar 16 2026 at 08:50)::laughing:
Matteo Capucci (he/him) (Mar 16 2026 at 08:50):tbh I found the companion so valuable exactly because it gives a lot of intuitions and a birdeye view of the subject
Steve Lack (Mar 16 2026 at 08:52):Jean-Baptiste Vienney said:
I will give a talk at Macquarie on Wednesday and I mentioned this strictification question in my slides. I wrote « Nathanael Arkor and Matteo Capucci gave me these two references: … I’m scared by Australian 2-category theory. Please help me! » I think Steven Lack will be in the audience (he is a professor there) so we’ll see what happens :joy:
Looking forward to your talk.
Jean-Baptiste Vienney (Mar 16 2026 at 17:39):Hi Steve, in fact I was mistaken and my talk is next week. So we’ll probably meet before