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Stream: learning: questions

Topic: examples of helpful or important 2-limits

David Egolf (Jul 19 2024 at 17:02):

I've recently become interested in "pseudo-pullbacks". That's because in #learning: questions > defining an internal category without using pullbacks? @Rémy Tuyéras suggested that we can use pseudo-pullbacks to help define a notion of "internal category" in certain categories that don't have pullbacks.

In that context, I recently learned that we can more generally consider "2-limits"! To me, this sounds like potentially an exciting mathematical world to explore. To begin with, I'm looking for specific examples of situations in which 2-limits can be helpful.

I am guessing that 2-limits can be used to generalize constructions involving limits to 2-categories, but I'd be interested in any particular examples where this is important! I'm also guessing that in some cases a related 2-limit exists when a desired limit doesn't necessarily exist, but again I'd be interested in any particular examples where this can be useful.

(When I use the word "2-limits", I mean a notion of limits in either a strict 2-category or in a bicategory).

John Baez (Jul 19 2024 at 17:38):

2-limits are important all over the place; it's hard to even know where to start! 2-limits are a generalization of limits, which are a generalization of products, which are a modern way of thinking about multiplication, so you're really asking "where does multiplication come up in 2-category theory".

I really like using pseudo-pullbacks for composing spans of categories. This is really important in Jim Dolan's approach to representation theory. But it comes up by a natural sequence of generalizations, like this:

1) We like relations between sets, and it's important to compose relations, like

X is a friend of a friend of mine iff $\exists$ Y (X is Y's friend and Y is my friend).

2) A span of sets is like a 'witnessed' relation, where we don't merely care about whether the relation holds (a truth value) but how many ways it holds (a set). E.g., X may be a friend of a friend of mine in a set of different ways, for different choices of Y. We compose spans of sets using pullback.

3) But sets are just watered-down categories. So we should look at spans of categories. These are composed using pseudopullback.

John Baez (Jul 19 2024 at 17:42):

This is a quick summary of a rather long mental process that started when people first began to use language to think about relations!

David Egolf (Jul 19 2024 at 17:45):

That is interesting! I'm surprised that pseudopullbacks are used to compose spans of categories. Spans of categories are spans in $\mathsf{Cat}$, if I'm understanding you correctly. And pullbacks exist in $\mathsf{Cat}$, I believe. So it's intriguing to me that pseudopullbacks are still useful even when pullbacks are available!

I also like the perspective of viewing a span of sets as a sort of generalized relation, where pairs of elements can be related in multiple distinct ways. Cool!

David Egolf (Jul 19 2024 at 18:16):

Upon reflection, I think I see why pullbacks are not so satisfactory in $\mathsf{Cat}$. We probably don't really want to require commutativity of the pullback square "on the nose". Instead, we probably prefer to do something like asking the pullback square to commute "up to isomorphism". I am guessing that in some cases the pullback of some categories will be "missing" parts that should really be there, if we weren't so picky about differences between isomorphic objects.

David Egolf (Jul 19 2024 at 18:26):

To illustrate that, here's a picture:
picture

Here we have a functor $F:A \to C$ and a functor $G:B \to C$. The image of the single object of $A$ under $F$ is isomorphic to, but not equal to, the image of the single object of $B$ under $G$.

David Egolf (Jul 19 2024 at 18:27):

So, if we form some pullback category having objects that are pairs $(a,b)$ with $a \in A_0$ and $b \in B_0$ such that $F(a) = G(b)$ - we'll just get an empty category. In this case, the pullback misses the fact that $F$ and $G$ really are in agreement (up to isomorphism) on where they send the single object provided to them.

David Egolf (Jul 19 2024 at 18:30):

Based on this example, I am guessing this: When one is working in a 2-category, one may prefer to use 2-limits, even if some corresponding limit of an underlying category exists!

John Baez (Jul 19 2024 at 20:14):

Exactly - you got it! The pseudopullback works better than the pullback in Cat, for exactly the reason you mention. If I have a composable pair of spans, and another composable pair of spans that's equivalent to the first (i.e. the categories involved are equivalent in a manner compatible with the maps between them), I want their composites to be equivalent. That won't work if we compose them using pullback! But it works for pseudopullback.

David Egolf (Jul 19 2024 at 23:31):

Inspired by this discusson, I guess that one might consider using a pseudopullback when determining the composable morphisms for a category internal to $\mathsf{Cat}$!

David Egolf (Jul 19 2024 at 23:36):

I notice that the nLab says that a double category is "an internal category in $\mathsf{Cat}$". However, it's unclear to me if the pullback or the pseudopullback (or some other version of a pullback?) is used in this context to determine the composable morphisms. (The information I seek is probably somewhere on that nLab page... I just find that page a bit intimidating currently!)

Kevin Carlson (Jul 19 2024 at 23:38):

In this case, it's the actual pullback that's meant. The pseudopullback would mean something like giving an operation that composes any two horizontal arrows $f:a\to b,g:b'\to c$ equipped furthermore with an isomorphism $b\cong b'.$ That would be really annoying and isn't the kind of thing that happens in examples.

Kevin Carlson (Jul 19 2024 at 23:40):

That said, an internal category in $\mathsf{Cat}$ is not actually the kind of double category we're usually interested in, as it has strictly associative composition operations for both kinds of arrows. You want to ask for something weaker, but not by making the pullback defining the composable pairs pseudo, rather, by making the composition operation itself on horizontal arrows only associative up to coherent isomorphism.

John Onstead (Jul 19 2024 at 23:52):

2-limits are much more diverse than ordinary limits, and have uses across category theory. One important example of 2-limit related to pseudo pullbacks is the notion of a comma object. In Cat, the comma objects are comma categories! When I was first learning about comma objects I got them confused with pseudo pullbacks since both involve cospans in Cat, and even now I find it difficult to fully disentangle these notions. There are also 2-limits known as lax limits; one example of a lax colimit is a Kleisli category and the Grothendieck construction can be defined in terms of a colax colimit.

Rémy Tuyéras (Jul 20 2024 at 00:05):

David Egolf said:

I've recently become interested in "pseudo-pullbacks". That's because in #learning: questions > defining an internal category without using pullbacks? Rémy Tuyéras suggested that we can use pseudo-pullbacks to help define a notion of "internal category" in certain categories that don't have pullbacks.

@David Egolf I am pointing out @John Onstead's comment because in our discussion, when the $\Gamma$ property includes all arrows, the $\Gamma$-pullback is a comma category

Rémy Tuyéras (Jul 20 2024 at 00:40):

Kevin Carlson said:

In this case, it's the actual pullback that's meant. The pseudopullback would mean something like giving an operation that composes any two horizontal arrows $f:a\to b,g:b'\to c$ equipped furthermore with an isomorphism $b\cong b'.$ That would be really annoying and isn't the kind of thing that happens in examples.

@Kevin Carlson I like that you are pointing out the practicality of this definition for pseudopullbacks. It makes me want to mention that relaxing isomorphisms to other kinds of "relationships" is more often used/practical (at least implicitly). For example, there are programming languages that can compose two algorithms typed as:

$\mathtt{string} \to \mathsf{int}$

and

$\mathtt{float} \to \mathtt{string}$

where the composition works because we have an inclusion $\mathsf{int} \hookrightarrow \mathtt{float}$.

Rémy Tuyéras (Jul 20 2024 at 00:45):

In general, one would use weak/pseudo/lax pullbacks to compose when there are intermediary processes involved in the composition that "we don't care to talk about."

Rémy Tuyéras (Jul 20 2024 at 00:46):

But these processes could potentially be as complicated as going through zig-zags of micro processes

David Egolf (Jul 20 2024 at 02:43):

Thanks everyone! That's really interesting! Maybe your comments above will finally inspire me to properly learn what a "comma category" is. I'd not heard of comma objects before, that I can recall, and those sound interesting too.

It is also very cool to hear that a Kleisli category can be viewed as an example of a "lax colimit" and that the Grothendieck construction is closely related to a "colax colimit"!

Tom Hirschowitz (Jul 20 2024 at 07:58):

A perhaps simpler example in the same vein is that the category of algebras for an endofunctor $F$ is the [[inserter]] of $F$ and the identity. I'm by far no expert on 2-limits, but from what I gather, an important use is that accessible/locally presentable categories are closed under a whole lot of 2-limits.

John Baez (Jul 20 2024 at 08:39):

David Egolf said:

Thanks everyone! That's really interesting! Maybe your comments above will finally inspire me to properly learn what a "comma category" is. I'd not heard of comma objects before, that I can recall, and those sound interesting too.

Once you understand comma categories, you can quickly understand comma objects: 'comma category' is a construction you can do in the 2-category $\mathbf{Cat}$, and if you mimic that construction in some other 2-category you call it a 'comma object' in that 2-category.

John Baez (Jul 20 2024 at 08:40):

The comma category construction is pretty simple: if you've got two functors $F: A \to X$, $G: B \to X$, then an object in the comma category is an object $a \in A$ and an object $b \in B$ equipped with a morphism $\alpha: F(a) \to G(b)$, and a morphism in the comma category is... the obvious thing.

John Baez (Jul 20 2024 at 08:42):

There are lots of interesting special cases of this extremely general construction.