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Stream: learning: questions

Topic: A certain 2-limit, what is it called?

fosco (Jan 03 2024 at 09:44):

I am considering the following situation: let $\cal K$ be a 2-category with all finite weighted limits. Then, an object $L$ is defined as follows, starting from 1-cells $f,g,y$:

$\begin{array}{ccccc}L&\to&g/y&\to&*\\\downarrow&&\downarrow&&\downarrow_y\\I(f,g)&\to&*&\underset g\to&*\\\downarrow&&\downarrow_g&&\\{*}&\underset f\to&*\end{array}$

the lower square is an inserter (so in particular, f,g are parallel 1-cells), the upper right square is a comma object, and the left upper square a strict 2-pullback. Does this particular shape of limit have a name?

David Michael Roberts (Jan 03 2024 at 10:46):

And I guess the asterisks are generic objects? Or a fixed object?

fosco (Jan 03 2024 at 10:49):

Generic objects, yes. Sorry I didn't y that out loud

fosco (Jan 03 2024 at 10:53):

You can assume that $f,g$ are endofunctors, because it's the case I am really interested in, but I don't think it changes much the general theory. I can certainly cook up a weight that presents $L$ as a limit "all at once", but it's not what I'm interested in.

What I would like is to know how I should think of L, in analogy with these wise sayings: an equalizer is "the greatest subspace where $f=g$" (think about Set). A filtered colimit is "a set of germs of functions" (think about sheaves).

Mike Shulman (Jan 03 2024 at 15:21):

Ah, okay. It's common for $\ast$ to denote a terminal object, so that threw me for a bit.

Mike Shulman (Jan 03 2024 at 15:51):

Since you wrote $I(f,g)$ and $g/y$, I assume the 2-cells in the squares are pointing to the upper right. I doubt this limit has a specific name. What sort of "wise saying" would you give to describe an ordinary comma object?

fosco (Jan 04 2024 at 10:01):

Mike Shulman said:

Ah, okay. It's common for $\ast$ to denote a terminal object, so that threw me for a bit.

I will make it clearer!

fosco (Jan 04 2024 at 10:03):

Mike Shulman said:

Since you wrote $I(f,g)$ and $g/y$, I assume the 2-cells in the squares are pointing to the upper right. I doubt this limit has a specific name. What sort of "wise saying" would you give to describe an ordinary comma object?

This is completely opinion-based, but I would say it's some sort of very very generalized bundle of a category over another.

Mike Shulman (Jan 04 2024 at 16:00):

Generally I think of a bundle as being a property of a map, rather than a way of constructing a span.