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Stream: theory: category theory

Topic: copointed endofunctors with this property

Matteo Capucci (he/him) (Jun 05 2023 at 08:22):

Hi y'all, as anyone ever stumbled upon an endofunctor $V:\cal C \to C$ with a copoint $p:V \Rightarrow 1_{\cal C}$ with the requirement that $Vp = pV$?

Matteo Capucci (he/him) (Jun 05 2023 at 08:26):

Actually $Vp \cong pV$ is enough, not that it makes much of a difference

Nathanael Arkor (Jun 05 2023 at 08:55):

These are called well-copointed endofunctors.

Matteo Capucci (he/him) (Jun 05 2023 at 15:15):

You're amazing Nathanel :D

John Baez (Jun 05 2023 at 16:48):

Hi y'all, as anyone ever stumbled upon an endofunctor $V:\cal C \to C$ with a copoint $p:V \Rightarrow 1_{\cal C}$ with the requirement that $Vp = pV$?

No. But if you replace Cat by an arbitrary 2-category this at least reminds me of something: in a 2-category with one object $\mathcal{C}$ and all morphisms and 2-morphisms invertible we often take a 1-morphism $V : \mathcal{C} \to \mathcal{C}$ and a 2-morphism $p: 1_{\mathcal{C}} \to 1_{\mathcal{C}}$ and form $p V p^{-1}$, which I sometimes call "whiskery conjugation" of $V$ by $p$.

Morgan Rogers (he/him) (Jun 05 2023 at 16:53):

@John Baez that "No." reads in context as a put-down of Nathanael :rolling_on_the_floor_laughing:

John Baez (Jun 05 2023 at 16:54):

Whoops! As usual I answer a question not seeing other previous questions. :grimacing: Will fix.

Matteo Capucci (he/him) (Jun 06 2023 at 07:30):

John Baez said:

Hi y'all, as anyone ever stumbled upon an endofunctor $V:\cal C \to C$ with a copoint $p:V \Rightarrow 1_{\cal C}$ with the requirement that $Vp = pV$?

No. But if you replace Cat by an arbitrary 2-category this at least reminds me of something: in a 2-category with one object $\mathcal{C}$ and all morphisms and 2-morphisms invertible we often take a 1-morphism $V : \mathcal{C} \to \mathcal{C}$ and a 2-morphism $p: 1_{\mathcal{C}} \to 1_{\mathcal{C}}$ and form $p V p^{-1}$, which I sometimes call "whiskery conjugation" of $V$ by $p$.

Uh I see... I'm not sure if that's relevant in my case but I'll keep it in mind :+1:🏻

John Baez (Jun 06 2023 at 19:06):

Yeah, it's not very relevant, just the best I could do. The more important thing is: how did you manage to change all the double dollar signs to single double dollar signs while quoting my comment, so that the LaTeX no longer works? That would seem to require actual effort. :upside_down:

Matteo Capucci (he/him) (Jun 07 2023 at 07:28):

Uh I didn't even notice

Matteo Capucci (he/him) (Jun 07 2023 at 07:28):

I used the quote & reply button on the mobile app, which is probably broken