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What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)

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When visiting Johns Hopkins this past April, I talked to Emily Riehl about this. It seemed like Smith's theorem should go through, with enough work (and this is the key input for the existence of localization in a combinatorial model category). I was planning to write up a short note to verify it, but haven't done so yet, and don't know yet what the precise hypotheses would be. If I recall correctly, Emily pointed out that any resulting paper would need a good application. Do you know of any model categories that are accessible but not combinatorial or cellular?

EDIT: I'm adding more details to sketch the result.

Tibor Beke's paper "Sheafifiable Homotopy Model Categories" presents Smith's theorem. Barwick has a paper that does too, and includes the application to left Bousfield localization. I now sketch a version of this theorem for accessible model categories. In the notation of Theorem 1.7 of Beke's paper, assume $M$ is a bicomplete category (not necessarily locally presentable), $W$ is a subcategory, $I$ is a category of morphisms (instead of a small set of morphisms), and assume:

  • c0: $W$ is closed under retracts and the 2-out-of-3 property
  • c1: inj$(I) \subset W$
  • c2: The class cof$(I)\cap W$ is closed under transfinite composition and pushout
  • c3: $W$ satisfies the solution set condition naturally at $I$ + more hypotheses.

The hypothesis c3 is not stated precisely above, but it must be strong enough to imply that the functors $M(i,-): W \to Set$ all satisfy the solution set condition, for every $i \in I$, coherently with respect to morphisms in $I$. This also implies $W$ is accessibly embedded.

The idea is to make c3 strong enough so that you can verify Beke's Lemma 1.8 (e.g. using the algebraic small object argument). Then, one should check that, if $M$ is an accessible model category, then the conditions c0-c3 are satisfied. Most of the work will be proving a variant of Smith's theorem in this context.

Now assume you had a version of Smith's theorem. Let $M$ be an accessible model category and $S$ a set of maps in $M$. In $L_S(M)$, you know that the pair (Cofibrations, Trivial Fibrations) is a weak factorization system, but you don't know the other one is. You can follow Barwick's paper to verify most of the model category axioms for $L_S(M)$ and the universal property of the localization. With the modified Smith theorem sketched above, you can construct a category $J$ that generates the other weak factorization system, simply by following Beke's proof and using the strengthened assumption c3. Once you have this weak factorization system, Barwick's paper completes the proof that $L_S(M)$ is the left Bousfield localization.

To summarize: I think the question the OP is asking is an open question, and I have sketched here the main result in a paper that would settle this open question. The reason no one has written up a paper on this is that it needs a stronger hypothesis c3 that would be very hard to check in practice, and there are seemingly no new examples this story would apply to. If the OP has an application in mind, and is interested to write up a paper about this, I'm happy to correspond via email.

EDIT 2: a couple of experts wrote to me to say this answer is probably too optimistic. A real attempt was made in this direction by Bourke, and anyone thinking about Smith's theorem for accessible model categories should read Bourke's paper. I'll come back to this in 6 months or so, when I have time, and will see if I can make the sketch any more precise. The OP and I are already in email contact.

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    $\begingroup$ By the way, the optimism in this post is entirely mine. If it turns out Smith's theorem doesn't go through, don't hold it against Emily. $\endgroup$ – David White Jul 17 at 12:54
  • $\begingroup$ We continue the discussion by email. $\endgroup$ – Philippe Gaucher Jul 17 at 15:19

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