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Stream: theory: category theory

Topic: 2-pushouts in 2-categories

Tim Hosgood (Jan 13 2023 at 11:09):

I'm interested in 2-pushouts in the 2-category of categories. The nLab has a page on [[2-pullback]] which has a very nice explicit description in §2. Equivalence of definitions. Three little questions:

1. Does $\mathsf{Cat}$ have all 2-pushouts? (Meta question: is there some place people use to look up properties of categories like this? The nLab is great for lots of things, but doesn't seem to have tables of properties like this)
2. In the universal property, it simply says that we have 2-isomorphisms $pu\cong v$ and $qu\cong w$, but is there a "natural" direction in which these actually point? As in, $pu\Rightarrow v$ or $v\Rightarrow pu$?
3. Does anybody know of a more stable reference (for the explicit description) that I can use in a citation?

Nathanael Arkor (Jan 13 2023 at 12:14):

1. Yes: Cat is cocomplete (even locally finitely presentable) as a 2-category, so has all 2-colimits.
2. I'm not sure whether there is a natural direction: others may have an opinion. My inclination would be to direct the 2-cells as $p u \Rightarrow v$ and $q u \Rightarrow w$, because then the universal property of the 2-pushout gives us 1-cells from the object of the 2-pushout, and 2-cells from its edges.
3. Kelly's Elementary observations on 2-categorical limits is a reasonable reference for (1).

Nathanael Arkor (Jan 13 2023 at 12:19):

Regarding the meta-question: my view is that the more useful thing to know is the various constructions that preserve cocompleteness. In the case of strict 2-colimits, this often follows from enriched category theory. So knowing various results for showing that a V-category is cocomplete is useful, e.g. that the category of algebras of a finitary V-monad on a cocomplete V-category is also cocomplete.

Nathanael Arkor (Jan 13 2023 at 12:22):

The nLab is great for lots of things, but doesn't seem to have tables of properties like this

In this case, I wouldn't expect the nLab to list every kind of 2-colimit [[Cat]] had, but I would expect it to mention that it's complete and cocomplete as a 2-category – but, looking at it now, it doesn't seem to! It would be great to add this to the page if you have the time :)

Tim Hosgood (Jan 13 2023 at 12:36):

all very helpful, thank you!

Mike Shulman (Jan 13 2023 at 15:40):

There have occasionally been abortive discussions on the nForum about putting together some kind of "database of categories" on or associated to the nLab, but they've never gone anywhere.