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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

Patrick Nicodemus (Apr 26 2025 at 17:19):

Great! Thank you, Bruno. I am interested here in generalizing the result that if $G : D \to C$ has a family of universal arrows $c \mapsto (d, \eta_c : c\to Gc)$ then $G$ has a left adjoint, to quasi-universal arrows and lax adjunctions.

Patrick Nicodemus (Apr 26 2025 at 17:20):

I'll come back and discuss this later if time permits. A source/reference would be helpful but i'm probably still going to write it up myself as older papers are harder to search and reference, and you run into terminological clashes.

Patrick Nicodemus (Apr 26 2025 at 17:45):

So, if $G : D\to C$ is a pseudofunctor between bicategories, and for every $c$ in $C$ the oplax comma category $c\downarrow G$ has a quasi-initial object, then we can prove that all this gives rise in a canonical way to a colax functor $C\to D$. Intuitively this seems like a kind of "left adjoint" but if the quasi-initial objects aren't unique up to equivalence then neither will be the associated colax functor. I think that one needs to further assume some conditions which amount to forcing the "left adjoint" to be also pseudo rather than colax.

Patrick Nicodemus (Apr 26 2025 at 17:47):

Somewhat annoyingly the condition I have in mind for uniqueness involves talking about all universal arrows into $G$, so it is really a property of $G$ rather than of any universal arrow into $G$. I find this less elegant because it doesn't give the same sense of a theorem that shows local data can be uniquely glued into global data.