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Stream: learning: questions

Topic: Monoidal functors to set

Martti Karvonen (Feb 04 2021 at 14:53):

So, everyone knows that the category of all functors ${C}\to\mathbf{Set}$ is very well-behaved. Assume now that $\mathbf{C}$ is equipped with a symmetric monoidal structure, and let us equip $\mathbf{Set}$ with the cartesian monoidal structure. How does the category of all symmetric monoidal functors ${C}\to\mathbf{Set}$ (and symmetric monoidal natural transformations) behave? I'm most curious about the lax monoidal case but happy to hear about the category of strong monoidal functors as well.

Mike Shulman (Feb 04 2021 at 15:40):

The lax monoidal functors $C\to \mathbf{Set}$ are the monoids in the Day convolution monoidal structure on $[C,\mathbf{Set}]$. So, as the category of monoids in a closed symmetric monoidal locally presentable category, they are locally presentable and otherwise fairly well-behaved.

Martti Karvonen (Feb 04 2021 at 16:09):

Nice, thanks!

Morgan Rogers (he/him) (Feb 04 2021 at 16:41):

That was wonderfully concise!

John Baez (Feb 04 2021 at 16:51):

(In Mike's answer C is assumed small.)

Martti Karvonen (Feb 04 2021 at 16:56):

I'm fine with assuming smallness.

John Baez (Feb 04 2021 at 16:57):

It was obvious, I just felt I had to say it in case any non-experts were listening.

Mike Shulman (Feb 04 2021 at 17:19):

The category of all functors $C\to \mathbf{Set}$ is not all that well-behaved either if $C$ is not small.