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Stream: learning: questions

Topic: "weak pushout" universal among pullbacks

Joe Moeller (Sep 01 2021 at 01:47):

https://arxiv.org/abs/1608.02664
In this paper, def 2.1, they define a "weak pushout" to be a pull back square which is initial amongst pullback squares that share the first two maps, as in the picture:
image.png

Joe Moeller (Sep 01 2021 at 01:47):

This is sorta funny to me. Has anybody seen a universal property like this before?

Reid Barton (Sep 23 2021 at 13:44):

I've seen this elsewhere but for the same purpose, namely to have a version of pushout which still works and gives the expected thing when you work in a category whose maps are just the monomorphisms of some original category.

Joe Moeller (Sep 23 2021 at 13:46):

Oh wow really? Could you share the paper? Was it also in the realm of rep theory?

Reid Barton (Sep 23 2021 at 14:02):

It's definition 5.3 of https://arxiv.org/abs/1811.08014v3 except I think the phrase "pullback square" (or square belonging to some chosen class of squares, which could be the pullback squares) is missing. But from context I believe that's what is intended.

Joe Moeller (Sep 23 2021 at 14:16):

Thanks! I'll check it out.

Spencer Breiner (Sep 23 2021 at 14:33):

Is this version of "weak" related to the usual meaning of weak (co)limits or are these orthogonal concepts?

Joe Moeller (Sep 23 2021 at 14:35):

I think they're different. The one in the paper I posted is existence and uniqueness, but among a restricted class of diagrams.

Spencer Breiner (Sep 23 2021 at 16:47):

I've been trying to think of a different name for this, but the best I've been able to come up with is a "mushout".

Reid Barton (Sep 23 2021 at 17:03):

The paper I linked to calls it a "restricted pushout", which at least doesn't have another usage, as far as I know

Ian Coley (Sep 23 2021 at 17:52):

This notion also comes up in a paper by Dyckerhoff-Kapranov on 'proto-exact categories' where there's an ambient (pointed) category that has coproducts, but they take a non-full subcategory which (among other things) loses the fold map A v A -> A. So they also make such a modification

Ian Coley (Sep 23 2021 at 17:53):

That might be the wrong citation, but it's definitely in a paper on proto-exact categories

Nathanael Arkor (Sep 23 2021 at 22:32):

Reid Barton said:

I've seen this elsewhere but for the same purpose, namely to have a version of pushout which still works and gives the expected thing when you work in a category whose maps are just the monomorphisms of some original category.

If I recall properly, this was the intention for introducing the "proxy pushouts" of Bumpus–Kocsis's Spined categories: generalizing tree-width beyond graphs, though there the solution looks different. I wonder if they're related at all.

Reid Barton (Nov 09 2021 at 15:15):

I recently encountered a kind of dual version of this.
The category Man of smooth manifolds and smooth maps doesn't have all pullbacks. But it does have pullbacks of submersions, and the pullback of a submersion is again a submersion. If we form the pullback of two submersions in Man, then we get a commutative square in the category Subm of smooth manifolds and submersions.
But this square is not a pullback square in Subm, because the map into the pullback given by the universal property in Man might not be a submersion when the original maps are submersions. For example, the diagonal map $M \to M \times M$ is never a submersion if $M$ has positive dimension.

Is there some way to characterize these squares in Subm (squares that are pullbacks in Man) purely in terms of Subm? Note that dualizing the condition that started this thread doesn't work--a pullback of submersions doesn't have to be a pushout, for example if $X \to Z$ and $Y \to Z$ have disjoint images which together do not make up all of $Z$.

Reid Barton (Nov 09 2021 at 15:16):

Alternatively, I wonder what kind of axiomatization of these "proxy pullbacks" would make sense (note for example they still satisfy the pullback pasting and cancellation laws, inherited from Man).

Reid Barton (Nov 09 2021 at 15:18):

The condition in the paper @Nathanael Arkor linked to is interesting, the existence and uniqueness of a morphism comparing two "proxy pullbacks" over the same base--I think that's satisfied in this Subm example.

Morgan Rogers (he/him) (Nov 09 2021 at 15:27):

Reid Barton said:

Alternatively, I wonder what kind of axiomatization of these "proxy pullbacks" would make sense (note for example they still satisfy the pullback pasting and cancellation laws, inherited from Man).

This conjures up a nice idea of being able to embed a given category equipped with proxy pullbacks faithfully into another category such that those squares become pullbacks

Morgan Rogers (he/him) (Nov 09 2021 at 15:42):

Besides the pasting property, you can observe that these are "sub-pullback squares", in the sense that given $f: M_1 \rightarrow N \leftarrow M_2 : g$ there is an injective function $\mathrm{Hom}_{Subm}(X,M_1 \times_N M_2) \hookrightarrow \{ (g',f') \in \mathrm{Hom}_{Subm}(X,M_1) \times \mathrm{Hom}_{Subm}(X,M_2) \mid f \circ g' = g \circ f' \}$ which is natural in $X$. I expect even this isn't quite enough, though, since one could have silly examples of sub-pullback squares.

John Baez (Nov 09 2021 at 16:22):

Reid Barton said:

The category Man of smooth manifolds and smooth maps doesn't have all pullbacks. But it does have pullbacks of submersions, and the pullback of a submersion is again a submersion. If we form the pullback of two submersions in Man, then we get a commutative square in the category Subm of smooth manifolds and submersions.

But this square is not a pullback square in Subm, because the map into the pullback given by the universal property in Man might not be a submersion when the original maps are submersions. For example, the diagonal map $M \to M \times M$ is never a submersion if $M$ has positive dimension.

Is there some way to characterize these squares in Subm (squares that are pullbacks in Man) purely in terms of Subm?

Adam Yassine and David Weisbart have done some work on this issue - but they decided it was too hard to characterize these squares purely in terms of Subm, so I can't actually your last question except to say "if you know one, you know more than us".

Their paper is here:

• Adam Yassine and David Weisbart, Constructing span categories from categories without pullbacks.

They come up with a formalism where you've got two categories and a functor between them, e.g. Subm $\to$ Man, and given any cospan in the first category its image in the second category has a pullback. (There's a bit more to it than this, but this is the basic idea.)

The reason they got into this was a project involving classical mechanics where we needed to compose spans of submersions:

• John Baez, Adam Yassine and David Weisbart, Open systems in classical mechanics.

We (and they) actually consider only surjective submersions, but that's probably not a big deal.

I imagine the whole setup could be polished and strengthened considerably. For example they construct a category of spans, but it's really a symmetric monoidal double category.

Joe Moeller (Jan 11 2022 at 21:44):

Is there any work done on a general theory of "restricted co/limits" where a cone is universal among some restricted class of cones over a diagram, rather than all of them?