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: learning: questions

Topic: notions of lax monoidal functors

Todd Trimble (Mar 14 2021 at 20:47):

I'm struggling to remember where I might have seen a notion of lax monoidal functor where the tensorator is an isomorphism but the unitor is merely lax. (I have a vague memory that the notion where the tensorator is lax but the unitor is pseudo is called a normal monoidal functor, but I'd like someone to weigh in there too.) Any ideas where such things have been studied?

John Baez (Mar 14 2021 at 20:51):

I haven't heard of "normal monoidal functors" in that sense; they're a cousin of "normal pseudofunctors", which are pseudofunctors from a 1-category to a 2-category where the unitor for composition of 1-morphism is the identity.

Todd Trimble (Mar 14 2021 at 21:01):

Thanks, John. Might one of your students know, e.g., Joe?

John Baez (Mar 14 2021 at 21:07):

Normal monoidal functors? I doubt it.

Nathanael Arkor (Mar 14 2021 at 21:47):

I have a vague memory that the notion where the tensorator is lax but the unitor is pseudo is called a normal monoidal functor, but I'd like someone to weigh in there too.

Steve Lack uses this terminology here, for instance.

John Baez (Mar 14 2021 at 22:00):

There's a nice result in there. Whenever a category has binary products and coproducts, it has a god-given natural transformation

$\delta_{x,y,z} : x \times (y + z) \to (x \times y) + (x \times z)$

If this is a natural isomorphism then we say the category is distributive. Claudio Pisani asked if the existence of any natural isomorphism

$\psi_{x,y,z} : x \times (y + z) \to (x \times y) + (x \times z)$

implies that $\delta$ is a natural isomorphism. And the answer is yes!

So this is not a counterexample - just a surprising 'better than you'd expect' result.

Nathanael Arkor (Mar 14 2021 at 22:13):

The cited result of Caccamo–Winskel's Limit Preservation from Naturality is quite nice too: it states a similar result for limit preservation (i.e. existence of a natural isomorphism, rather than invertibility of the canonical isomorphism, is sufficient).

Todd Trimble (Mar 15 2021 at 01:25):

Yes, this is all nice; thanks. Thanks, Nathanael, for the mention of the Lack paper. My main curiosity at the moment is about pseudo tensorator, lax unitor -- the stuff about normal monoidal functors was more of a side curiosity.

Paolo Perrone (Mar 15 2021 at 17:42):

The "normal" lax monoidal functors are studied for example here: Monoidal functors, species and Hopf algebras.