The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to avoid any tameness assumption but I can't avoid a couple of results about localizations.

Thus, the content of this question is: (if true) are the following two statements already in the literature?

Hope 1: Let $\mathsf{M}$ and $\mathsf{N}$ be model categories and $\mathsf{L}: \mathsf{M} \leftrightarrows \mathsf{N}: \mathsf{R}$ be Quillen adjunction which specifies $\mathsf{N}$ as a reflective subcategory of $\mathsf{M}$, then $\mathsf{N}$ is Quillen equivalent to a Bousfield localization of $\mathsf{M}$, where the new weak equivalences $\bar{\mathcal{W}}$ are $\mathcal{W}_{\mathsf{M}} \cup \mathsf{L}^{-1}(\mathcal{W}_{\mathsf{N}})$.

This should be seen as a technical improvement of Rem. 3.8 on the nlab page about idempotent monads.

Hope 2: Let $\mathsf{M}$ and $\mathsf{T}$ be an idempotent monad over $\mathsf{M}$, then there exists a model structure on $\mathsf{Alg}(\mathsf{T})$ and a Quillen adjunction $\mathsf{L}: \mathsf{M} \leftrightarrows \mathsf{Alg}(\mathsf{T}): \mathsf{R}$ such that $\mathsf{Alg}(\mathsf{T})$ is Quillen equivalent to a Bousfield localization of $\mathsf{M}$.

This would be a variant of Prop. 3.8 in the lecture notes by Urs Schreiber where the model structure is not assumed to be right proper and the notion of idempotent monad is strictified. In Def. 3.3, while I really like that $\mathsf{T}$ does not have to be a monad, I find quite unnatural the condition (3) and the assumption that the structure is right proper in Prop. 3.8.

It is fine to assume that $\mathsf{M}$ is right proper if needed, but I cannot expect the adjunction to respect it.

  • $\begingroup$ Just to be clear: are you interested in proofs of these statements, or you already know how to prove them and are merely interested in references? Also, in Hope 1 $\bar W$ should probably be closed under composition and 2-out-of-3 first. $\endgroup$ – Dmitri Pavlov Nov 13 '19 at 15:12
  • $\begingroup$ Thanks for the question, I am interested in both proofs and statements. $\endgroup$ – Ivan Di Liberti Nov 13 '19 at 15:28

So far, the best results that I managed to find in the direction of my questions appear both in some paper of Boris Chorny and his collaborators.

1st. Theorem, A non-functorial Bousfield-Friedlander localization,

This result appears as A.8 in the paper below, it is a bit too long to be cast in this answer, in a nutshell it provides a version of the Hope 2 when $\mathsf{T}$ is not even a functor.

Duality and small functors, B. Chorny and G. Biedermann. Algebr. Geom. Topol. 15 (2015) 2609-2657

2nd. Theorem, A generalization of Smith's theorem,

Thm 3.14 (C.R.) Let $\mathsf{M}$ a nice, proper, simplicial class-combinatorial model category and $\mathsf{T} : \mathsf{M} \to \mathsf{M}$ a strongly class-accessible homotopy idempotent functor preserving weak equivalences. Suppose additionally, that pullbacks of $\mathsf{T}$-equivalences along $\mathsf{T}$-fibrations are $\mathsf{T}$-equivalences. Then the $\mathsf{T}$-localization exists and is class-combinatorial.

The main feature of this theorem is that the localization is class-combinatorial.

This result appears in,

Class combinatorial model categories, B. Chorny and J. Rosický, Homology Homotopy Appl. 14 (2012), no. 1, 263--280

I recommend the reader not to be treated by the use of the world accessible and locally presentable. Class-accessibility is a very strong weakening of the notion of accessibility. On one hand, it escapes the world of categories with a dense (small) generator, on the other, it still allows to build on the technical power of the small object argument.

Another paper that should be mentioned is

Locally class-presentable and class-accessible categories, B. Chorny and J. Rosický, J. Pure Appl. Alg. 216 (2012), 2113-2125.

where the general theory of class-accessibility and class-local presentability is discussed.


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