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Stream: theory: category theory

Topic: lax/weak monoidal categories

Nicolas Blanco (Oct 03 2022 at 12:17):

In monoidal category theory, it is sometimes easier to work with an unbiased notion. It is the case for example when establishing the correspondence (2-equivalence really) between monoidal categories and representable multicategories. From there, it is not a big step to direct things and get the notion of lax/weak monoidal category. The correspondence mentioned above can be extended to a correspondence (and 2-equivalence) between oplax monoidal categories and weak representable multicategories, where the tensor products have a weaker universal properties: any multimap with domain $\Gamma$ can be uniquely factorise through the universal multimap $\Gamma \to \otimes \Gamma$ followed by a unary multimap.
My first question will be: Do you know any examples of lax/oplax monoidal categories? It is even better if they come from other mathematical areas than category theory.

Now let move from multicategories to polycategories. Similarly to above, when establishing the correspondence between linearly distributive categories and two-tensor polycategories, it might be easier to work with an unbiased notion of linearly distributive categories. And from there, going to a lax/weak version is not much difficult. By unraveling what is needed for this correspondence to exist we get the following notion of lax linearly distributive category.

A lax linearly distributive category is a category $\mathcal{C}$ with:

• An oplax monoidal structure $(\mathcal{C}, \otimes_n)$
• A lax monoidal structure $(\mathcal{C}, ⅋_n)$
• distributivity laws, i.e. natural transformations $\otimes(\Gamma_1,⅋(\Delta_1,A,\Delta_2),\Gamma_2) \to ⅋(\Delta_1,\otimes(\Gamma_1,A,\Gamma_2),\Delta_2)$ with the restriction that $\Gamma_i = \emptyset$ or $\Delta_i = \emptyset$ in the non-symmetric case

subject to some coherence laws.
A lax ldc is called normal if the natural transformations $\otimes_1 - \Rightarrow -$ and $- \Rightarrow ⅋_1 -$ given by the monoidal structures are invertible.
There is a correspondence between lax normal linearly distributive categories and weak two-tensor polycategories (if my calculations are correct of course).

Second question: Can you think of any candidate that could be an example of a lax linearly distributive category?

Mike Shulman (Oct 03 2022 at 15:42):

There's an example of an oplax double category in section B.11 of my paper A unified framework for generalized multicategories with Geoff Cruttwell. And some lax monoidal structures are used to generalize the Gray tensor product in Batanin-Cisinsiki-Weber Multitensor lifting and strictly unital higher category theory. But those are both fairly pure category theory.

Matteo Capucci (he/him) (Oct 03 2022 at 18:00):

In this paper they prove cartesian differential categories are categories enriched in a 'skew monoidal' category given by 'twisting' the usual tensor of modules over a rig with a monoidal comonad. IIRC skew monoidal and lax monoidal are synonyms?

Mike Shulman (Oct 03 2022 at 18:11):

No.

Nathanael Arkor (Oct 03 2022 at 18:11):

There's a helpful comparison between the concepts on [[lax monoidal category]].

Nicolas Blanco (Oct 04 2022 at 13:13):

Thanks @Mike Shulman
I will look at these examples. Am I right in assuming that an oplax double category is a virtual double category with weak composites, i.e. the unique factorisation property only holds through unary 2-cell.

Nicolas Blanco (Oct 04 2022 at 13:14):

Thanks @Matteo Capucci (he/him) :) But skew monoidal and lax monoidal are indeed different way of relaxing monoidality. And thanks @Nathanael Arkor for the link to the nlab page!

Mike Shulman (Oct 04 2022 at 15:52):

Yes, that's right.