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Stream: theory: category theory

Topic: Infinitary distributive monoidal category

Jean-Baptiste Vienney (Aug 03 2023 at 19:33):

On the nlab, they say this:
Screenshot-2023-08-03-at-3.29.21-PM.png
Is there a paper where the precise coherence conditions are described in the infinitary case? I would even like a bit more, I'm interested by $\mathbb{N}$-distributive symmetric monoidal categories.

Nathanael Arkor (Aug 03 2023 at 21:02):

There are no coherence conditions. In a monoidal category with infinite coproducts, we have canonical morphisms $[X \otimes \iota_{Y_i}]_i \colon \coprod_i (X \otimes Y_i) \to X \otimes (\coprod_i Y)$ (and similarly for tensoring with a coproduct on the left) and we ask that these morphisms are all invertible.

Jean-Baptiste Vienney (Aug 03 2023 at 21:10):

Oh I see, so the coherences that I'm thinking of, such as the 24 diagrams in the definition of symmetric bimonoidal categories that are listed in "Coherence for distributivity" by Laplaza, are thus facts to deduce from the universal properties of coproducts + the fact that this canonical morphisms are isomorphisms, and not part of the definition. It makes sense, thanks.

Jean-Baptiste Vienney (Aug 03 2023 at 21:15):

It would still be useful to define the coherence conditions for an infinitary symmetric bimonoidal category ie. one with a functor $\oplus_{i}:\mathcal{C}^I \rightarrow \mathcal{C}$ and a structure of symmetric monoidal category on $\otimes$. And they should be verified, when $\oplus_{i}$ is a coproduct, as a proposition.

Jean-Baptiste Vienney (Aug 03 2023 at 21:17):

But we will need the definition of monoidal category with a tensor product which takes a family indexed by any set $I$ in input first haha

Jean-Baptiste Vienney (Aug 03 2023 at 21:18):

I don't know examples of this other than when this infinite tensor product is given by product or coproduct, but maybe it exists...

Nathanael Arkor (Aug 03 2023 at 22:27):

Jean-Baptiste Vienney said:

But we will need the definition of monoidal category with a tensor product which takes a family indexed by any set $I$ in input first haha

There's a definition (both symmetric and nonsymmetric variants) in the paper Real sets of Janelidze and Street.

Claudio Pisani (Aug 04 2023 at 15:55):

An alternative definition of infinitary symmetric monoidal category is in my paper Operads as double functors
(as sketched in the introduction); namely, as product preserving lax double functors
$(\mathbb{P}bSet)^{op} → \mathbb{S}et$, from the double category of pullback squares in sets to the double category of spans in sets.
Infinitary commutative monoids arise as product preserving double functors $(\mathbb{P}bSet)^{op} → \mathbb{S}q(Set)$
to the double category of squares in sets.