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Stream: theory: category theory

Topic: Small object argument in a cofibration category?

Kevin Arlin (Sep 27 2023 at 22:59):

I want to construct a weak factorization system using the small object argument in something like a homotopy cocomplete cofibration category, though I think I don't need to specify any weak equivalences. Basically I have a category $\mathcal A$ with a class of cofibrations $\mathcal C,$ I don't necessarily have all pushouts but I can pushout cofibrations along any map, I have small coproducts and coproducts of cofibrations are cofibrations, and I have transfinite compositions of cofibrations.

It seems pretty obvious that if I have a small class of cofibrations between small objects in such a setting, then I can run the small object argument exactly as usual. But the small object argument is (almost?) always described for categories such as model categories that are actually cocomplete. Does anybody know whether there's a reference for this mild generalization?

Tom Hirschowitz (Oct 10 2023 at 09:19):

@Kevin Arlin I was interested when you asked the question, but now I kind of need the result :innocent:
Did you get any useful feedback or find anything on your own? Or even write up the result yourself?

Reid Barton (Oct 10 2023 at 17:17):

I'm curious what an application would be where you don't already have an "enveloping" cocomplete category with a WFS at hand, in which $\mathcal{A}$ is the full subcategory of "cofibrant" objects.

Kevin Arlin (Oct 10 2023 at 19:05):

I'm actually thinking of $\mathcal A$ as a wide subcategory, Reid. That said, the key examples I have in mind are where $\mathcal A$ is a 2-category with flexible limits and colimits. These are omnipresent in 2-category theory, starting with the 2-category of categories with finite products and that of monoidal categories with strong monoidal functors, because whenever $\mathcal K$ is a reasonable 2-category and $T$ is a reasonable 2-monad on it, the Eilenberg-Moore 2-category of $T$ -algebras and pseudomorphisms has these properties.

Kevin Arlin (Oct 10 2023 at 19:05):

And in cases like this I really don't think there's any reasonable envelopment like you're thinking of in cases more like CW-complexes vs. nice-category topological spaces.

Kevin Arlin (Oct 10 2023 at 19:06):

Interested to know what your application is, Tom! I'm planning to just write it up and have it in an appendix of our paper in prep, I'll send you something privately.

Kevin Arlin (Oct 10 2023 at 19:08):

(Oh, and in my application, $\mathcal C$ is the normal isocofibrations, which is usually the same as the 1-morphisms sent to isofibrations in $\mathbf{Cat}$ by representables.)

Tom Hirschowitz (Oct 11 2023 at 05:54):

Oh, so you need the enriched variant of the small object argument? Btw, did you have in mind the original Quillen version, or the more algebraic one by @Richard Garner, John Bourke, @Emily Riehl...?

Tom Hirschowitz (Oct 11 2023 at 06:31):

Anyway, my application is in the context of familial functors, following my paper with Richard. Familial functors enjoy a spectrum-exponent representation.
This means that familial functors $𝒞 → \widehat{𝔻}$ are equivalent to pairs of a spectrum presheaf $S ∈ \widehat{𝔻}$ , and an exponent functor $E\colon 𝐲/S →𝒞$ , where $𝐲/S$ denotes the category of elements of $S$ . The original functor is recovered as $C ↦ c ↦ ∑_{x ∈ S(c)} 𝒞(E(c,x),C)$ .

We have a notion of analytic functor which generalises familial functors, essentially by replacing $𝒞$ with a category of orbits $𝒪(𝒞)$ . An object of this category is a pair $(C,G)$ of an object of $𝒞$ and a subgroup $G$ of the automorphism group of $C$ . (This is a generalisation to presheaf categories of Joyal's notion of analytic functors.) The idea is that analytic functors enable some limited form of quotienting. E.g., the free commutative monoid monad is analytic, but not familial. Its spectrum is the set $ℕ$ , and its exponent maps any $n$ to the $n$ -element set, with its full automorphism group.

The problem is that analytic functors behave badly, e.g., they are not even closed under composition. Richard came up with a slightly less general notion, called cellular functor, which fixes some issues — though not all. The core issue is that $𝒪(𝒞)$ does not have colimits, even the most basic ones like pushouts. The rough idea is to restrict to a well-behaved subcategory, say $ℐ$ of cell complexes, and take exponents $𝐲/S → ℐ$ . But in fact we need to take it a bit further. In the paper this is done with additional technical conditions, but it may be viewed as equipping also $\widehat{𝔻}$ with cell complexes, say $𝒥$ , and taking well-behaved exponents $𝒥/S → ℐ$ . The point now is that cell complexes are defined in a rather ad hoc way in the paper, so, my student Yoann Barszezak and I would like to view them as cofibrant objects in $𝒪(𝒞)$ , which involves setting up a weak factorisation system. As I said, $𝒪(𝒞)$ does not have pushouts, but it does have pushouts along our candidate generating cofibrations (which was in fact the whole point of cell complexes from the beginning), and we think it has transfinite compositions.

@Reid Barton, it might be a good idea to (fully) embed $𝒪(𝒞)$ into $[𝒞,\mathbf{Set}]$ and work there, but there will probably be a few things that we won't know how to do. E.g., a notion that is crucial to the theory is that of vertical morphism in $𝒪(𝒞)$ , which is essentially one that merely modifies automorphisms (= the underlying morphism is an iso). I don't see offhand how to extend this to $[𝒞,\mathbf{Set}]$ .

Of course we'll be grateful for constructive comments and suggestions!

Kevin Arlin (Oct 11 2023 at 18:12):

@Tom Hirschowitz I'm hoping to just run the small object argument on the underlying categories, but I'm not completely sure yet whether that'll go through, after thinking more about the colimits in these 2-categories.

Kevin Arlin (Oct 11 2023 at 18:14):

But the basic overlap of being in a bad 1-category that does have pushouts along a distinguished class of maps is still there. Looks like an interesting case.

Kevin Arlin (Oct 11 2023 at 19:07):

Ah, and I'm thinking mainly about the old-fashioned small object argument but am certainly interested in thinking about the algebraic one, at least in principle.

Mike Shulman (Oct 11 2023 at 19:18):

You can, of course, always embed a given category into a cocomplete one universally, via its Yoneda embedding. That embedding won't preserve the existing colimits, but it does if you restrict the Yoneda embedding to land in presheaves that themselves preserve those colimits (qua functors to Set).

Reid Barton (Oct 16 2023 at 19:47):

Thanks for these explanations.

@Kevin Arlin 's example is similar to things I have thought about before, so maybe I can say something useful. Let's take the example of the 2-category of (categories with finite products, functors that preserve finite products [in the usual up-to-iso] sense, all natural transformations). It sounds like your approach is to keep these 1-morphisms, get rid of the 2-morphisms, and work with the 1-category that results and whatever (co)limits it still has.

Two other possible approaches would be:

Take categories with chosen finite products, and functors that strictly preserve the chosen finite products. This is not very nice from the 2-categorical viewpoint, but on the other it does produce a 1-category that is exactly the category of models of an essentially algebraic theory, hence is locally presentable and in particular has all colimits and limits.

In general these (co)limits will not be homotopically meaningful (i.e., equivalence-invariant). We can try to address using a model structure in which, for example, the cofibrant objects are the ones where the chosen finite products are "freely adjoined". The idea is that, when we consider functors out of a cofibrant category-equipped-with-chosen-finite-products, the category of strictly-product-preserving functors should be equivalent to the category of functors that are product-preserving in the usual up-to-iso sense. If we don't care about a full model category structure, we could also restrict attention to the cofibrant categories and make them into a cofibration category.

This approach is maybe a bit awkward if you would prefer to be working with categories up to equivalence (e.g., if one of the categories you would like to consider is "the" category of sets), since it requires distinguishing equivalent categories with non-isomorphic chosen-finite-product structure. For example, if $1$ is the chosen terminal object, the chosen product $1 \times 1$ might or not be equal as an object to $1$ , (and, therefore, in a cofibrant category they will be forced to be non-equal).
Work with the "correct" 2-categorical (co)limit notions through out, so that everything is equivalence-invariant. If you only need some specific colimits (say initial objects, pushouts, sequential colimits, and maybe some tensors), then this needn't be very technical. On the other hand, you would have to check that whatever you want to do with a cofibration category also works in a "cofibration 2-category".

With the suggested approach of using the 1-category of up-to-iso product-preserving functors, I find the situation a bit confusing. It seems that this approach works better for limits than for colimits. For example, this approach is used in https://arxiv.org/abs/1411.0303 to build a fibration category of cofibration categories, in which the fibrations are (among other conditions) isofibrations. But for colimits it seems problematic. For example, there cannot be an initial object in this category, since whatever it is, it should admit a unique product-preserving functor to any other category $X$ , yet we could replace $X$ by $X \times S$ where $S$ is a contractible category with multiple objects, and then there cannot also be a unique product-preserving functor to $X \times S$ .

However, this setup also depends delicately on the exact 2-category we started with, and discarded the 2-morphisms of. It could be that we can replace it by a (bi)equivalent 2-category with more success. For instance, this happens with (categories and left adjoints), because we could work instead with the opposite of the category formed by the right adjoints. I don't have a good sense for when this kind of strategy should succeed.

Kevin Arlin (Oct 16 2023 at 20:32):

Thanks for the thoughts, Reid. Switching to the strict-morphisms category probably doesn't help if I insist on getting an orthogonal factorization system, since precisely as you say the underlying categories are far from equivalent. The hope is to do something closer to 2. I don't really need much about 2-cofibration categories because I'm not localizing or anything. A third option might be to give up on the strictly orthogonal factorization and instead look for one with unique-up-to-iso factorizations. But this seems annoying.

Kevin Arlin (Oct 16 2023 at 20:39):

The colimits that are present in this kind of situation are a case of what Bourke et al call "shrinkable colimits" here: https://arxiv.org/pdf/2006.07843.pdf

Specifically these 2-categories, if I understand correctly, have all weighted colimits for which there is merely a surjective equivalence from the maps from the colimit to the category of cocones/cocylinders. This is enough to get all weak colimits in the underlying category, but that's pretty much it. I didn't understand this till last Friday and I'm now less confident what one might be able to get away with here.

Kevin Arlin (Oct 16 2023 at 21:25):

Hm, actually, this afternoon it's looking to me like option #1 might get me somewhere. There's such a nice 2-equivalence between the strict-morphism 2-category and the pseudo-morphism one that I seem to be able to transfer orthogonal factorization systems across on the nose.