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: practice: terminology & notation

Topic: Lax initial object

Patrick Nicodemus (Apr 23 2025 at 22:24):

Let C be a 2-category, and let x be a distinguished object in C.

What do you call it when C(x,y) has an initial object for all y?

I almost want to say "lax initial" but it doesn't seem to be a lax limit.

Amar Hadzihasanovic (Apr 23 2025 at 22:35):

This with the additional constraint that $\mathrm{id}_x$ is initial in $C(x, x)$ is dual to what Johnson and Yau (Definition 7.2.3) call an inc-lax terminal object (inc stands for "initial component").

Amar Hadzihasanovic (Apr 23 2025 at 22:36):

(So following their language it should be called inc-lax initial object)

Patrick Nicodemus (Apr 23 2025 at 22:39):

Oh, okay. Yeah. I'm realizing this is not unique up to equivalence.

Bruno Gavranović (Apr 25 2025 at 13:01):

There was a thread about this question ~2 or so years ago, here, and the conclusion that it indeed clashes with the terminology of a lax limit.

In my thesis (Appendix E) I used the term quasi-initial, and unpacked this particular construction in detail