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Stream: learning: questions

Topic: The category of locally presentable categories

James Deikun (Oct 29 2025 at 13:43):

For each particular $\kappa$, the category of locally $\kappa$-presentable categories and $\kappa$-accessible adjunctions is locally $\kappa$-presentable. However, the category of locally presentable categories and accessible adjunctions is not locally $\kappa$-presentable for any particular $\kappa$, hence not locally presentable. Aside from the existence of a small dense generator, what notable properties of locally presentable categories does it lose?

Also, what kind of properties do the inclusion functors induced by the ordering $\kappa \le \lambda$ of regular cardinals have?

Mike Shulman (Oct 30 2025 at 00:32):

Are you using "category" to mean "$(\infty,1)$-category"?

James Deikun (Oct 30 2025 at 00:35):

I wasn't particularly, but I'd be interested in either. (Or if something I said above is only true of $(\infty,1)$-categories, then I'd been misinformed and would be glad to know about it.)

Mike Shulman (Oct 30 2025 at 00:37):

There isn't a very meaningful 1-category whose objects are locally $\kappa$-presentable 1-categories. You could write one down, but it won't be very well-behaved, and certainly won't be locally presentable, until you consider it as a 2-category.

David Michael Roberts (Oct 30 2025 at 03:13):

I guess James might have been inspired by Scholze's recent https://people.mpim-bonn.mpg.de/scholze/Gestalten.pdf or comments he has already made publicly

Screenshot 2025-10-30 at 1.41.42 PM.png

He points to a proposal of Aoki https://arxiv.org/abs/2510.13503, given in Scholze's Definition 1.15

James Deikun (Oct 30 2025 at 05:22):

Yes, it seems like this was the original source which inspired the question. Apparently some important details were distorted in the process of (oral) transmission.

Alexander Campbell (Oct 30 2025 at 23:58):

In the notes for the second lecture, Scholze cites a discussion, begun by @Tim Campion, from this very forum as a reference for this fact: #theory: category theory > this category "is an object of itself"! @ 💬

David Michael Roberts (Oct 31 2025 at 00:49):

I would love to see Mochizuki's reaction to the notes from this specific lecture.

Kevin Carlson (Oct 31 2025 at 22:20):

What is the connection to Mochizuki?