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Stream: theory: category theory

Topic: microcosm principle for actegories

Bruno Gavranović (Apr 02 2021 at 18:30):

Hi, @Jules Hedges here writing from Bruno's laptop trying to survive on a Croatian keyboard. Let $M$ be a monoidal category. There's a 2-category $M\mathrm{-Act}$ whose morphisms are $M$-actegories and morphisms are functors that preserve the action. Is there a natural way to view $M\mathrm{-Act}$ itself as a (large) $M$-actegory?

The first thought is that for an $M$-actegory $\mathcal C$ and an object $m : M$, define the $M$-actegory $m \cdot \mathcal C$ to have the same underlying category $\mathcal C$, and the action given by $m' \cdot x = m' \cdot (m \cdot x) = (m' \otimes m) \cdot x$. Unfortunately this appears to not satisfy the condition to define an actegory. Does anyone know anything about this?

Mike Shulman (Apr 02 2021 at 19:00):

Not exactly an answer to your question, but I would say the application of the "microcosm principle" to actegories is that if $C$ is a category with an $M$-action, for $M$ a monoidal category, then one can define internally the notion of an object $x\in C$ with an $m$-action for a monoid object $m\in M$. One step up, the notion of a category $C$ with an action by a monoidal category $M$ would be defined internally to the "2-actegory" of $\rm Cat$ with an action by the monoidal category $\rm Cat$.

Jade Master (Apr 02 2021 at 20:21):

I think that you need the objects of mC to be the products of m with the objects of C instead of just the objects of C.

Jade Master (Apr 02 2021 at 20:23):

Similarly for the morphisms. It's morally the same as having objects and morphisms the same as C but now it respects the unit. Is that the problem you were having @Jules Hedges

Jules Hedges (Apr 03 2021 at 09:50):

Mike Shulman said:

One step up, the notion of a category $C$ with an action by a monoidal category $M$ would be defined internally to the "2-actegory" of $\rm Cat$ with an action by the monoidal category $\rm Cat$.

I think this is the same thing that Bruno came up with and we spent several hours working with, which is reassuring. We kept saying "Cat-actegories", which I find a funny thing to say aloud

Jules Hedges (Apr 03 2021 at 09:53):

Jade Master said:

Similarly for the morphisms. It's morally the same as having objects and morphisms the same as C but now it respects the unit. Is that the problem you were having Jules Hedges

Hm, I don't understand this

Jules Hedges (Apr 03 2021 at 09:56):

I should write why the proposed action I wrote in the first post isn't actually an action. Let me write $\cdot$ for the action of $M$ on $\mathcal C$, and $\bullet_m$ for the proposed action of $M$ on $m \cdot \mathcal C$, ie. $m' \bullet_m x = m' \cdot m \cdot x$

Jules Hedges (Apr 03 2021 at 09:58):

Then
$(m_1 \otimes m_2) \bullet_m x = m_1 \cdot m_2 \cdot m \cdot x$
but
$m_1 \bullet_m (m_2 \bullet_m x) = m_1 \cdot m \cdot m_2 \cdot m \cdot x$

Jules Hedges (Apr 03 2021 at 09:59):

It still might be possible to salvage this idea by defining $m \cdot \mathcal C$ more cleverly