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Stream: theory: category theory

Topic: where can i look for a laxator

Matteo Capucci (he/him) (Apr 13 2021 at 16:04):

Say I have a functor $F: C \to D$ between monoidal categories. Is there some 'standard trick' or 'obvious way' to look for a lax monoidal structure on $F$? Do I have to guess and check?
On a separate note, are laxators structures or properties?

Joe Moeller (Apr 13 2021 at 16:11):

I think it's a very case-by-case thing. One naturally occurring example is when the monoidal structure is cocartesian in both categories. Every functor is lax monoidal when both categories have finite coproducts.

Jade Master (Apr 13 2021 at 16:18):

And oplax monoidal when both categories have products :)

Jade Master (Apr 13 2021 at 16:23):

Also the laxator of a composite factors into laxators on the components...so factoring your functors can help.

Reid Barton (Apr 13 2021 at 16:30):

For a suitable choice of $C$ this question should reduce to: suppose $X$ is an object of a monoidal category $D$; how can I find a monoid structure on $X$? So it's hard to imagine there is much to say about this problem in general.

Jade Master (Apr 13 2021 at 16:40):

(deleted)

Cody Roux (Apr 13 2021 at 16:41):

Walgreens? (sorry)

Fabrizio Genovese (Apr 13 2021 at 16:59):

Cody Roux said:

Walgreens? (sorry)

Damn, I've been too slow.

John Baez (Apr 13 2021 at 17:53):

I used to call them "laxatives" until my students rebelled and one of them advocated "laxator". Does anyone know the first paper in which "laxator" appeared? Or was it the nLab?

John Baez (Apr 13 2021 at 17:54):

Older papers tend to say something like "the lax monoidal structure" or something wishy-washy like that.

John Baez (Apr 13 2021 at 17:56):

oplax

Matteo Capucci (he/him) (Apr 13 2021 at 18:49):

Cody Roux said:

Walgreens? (sorry)

Fair enough :laughing:

Matteo Capucci (he/him) (Apr 13 2021 at 18:49):

Thanks for the replies, @Jade Master, @Joe Moeller, @Reid Barton

Matteo Capucci (he/him) (Apr 13 2021 at 18:50):

What about my second question? Is being lax monoidal a property or a structure? :thinking:

Jade Master (Apr 13 2021 at 18:57):

A structure I think. A functor F is lax monoidal if it is equipped with a natural transformation $L: F(-) \otimes F(-) \to F(-\otimes -)$ satisfying appropriate axioms.

Jade Master (Apr 13 2021 at 19:00):

So we have the structure of a laxator imposed on the functor F.

Mike Shulman (Apr 13 2021 at 19:01):

Reid's comment gives a proof that it must be a structure: if $C=1$ then a functor $F:C\to D$ is just an object of $D$, and a lax monoidal structure on it is a monoid structure on that object, which is certainly a structure and not just a property.