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: When a Kan = Comma?

Beppe Metere (Jul 22 2026 at 08:22):

Yes, for instance: let $C$ be any category and

$$U=dom:C^2\to C, \qquad U'=cod:C^2\to C.$$

One has

$$U\vdash F: C\to C^2, \qquad A\mapsto 1_A,$$

so that $1_C=Ran_{U}(U').$

On the other hand

$$C^2\cong (1_C\downarrow 1_C).$$

Other examples can be produced similarly...

Giuseppe Leoncini (Jul 22 2026 at 14:59):

This reminds of proposition 8.4.10 (ii) in Riehl&Verity's "Elements" book. So I guess, given $U \colon E \to S$ and $U'\colon E \to T$, a general criterion should be: the canonical map $K:E\to (({\rm Ran}_U U')\downarrow 1_T)$ is an equivalence IFF $T \xleftarrow{U'} E \xrightarrow{U} S$ is a profunctor (a two sided discrete fibration) such that $U$ admits a left adjoint.

Beppe Metere (Jul 22 2026 at 17:02):

Yes! That is a representable profunctor. Thank you for the reference!

Beppe Metere (Jul 22 2026 at 18:17):

Maybe this conversation could be moved to "learning: questions" :smiling_face:

John Baez (Jul 23 2026 at 12:20):

I did it. I didn't know I had the power. You too had the power.

On a web browser, on goes up to the headline "learning: questions > When a Kan = Comma?" and clicks on the 3 dots at right.

Beppe Metere (Jul 23 2026 at 13:58):

John Baez said:

I did it. I didn't know I had the power. You too had the power.

I didn't know either I had this power! Now I know. Thank you John.