Localizing Model Structures

Question

I came along the following question while trying to understand and apply some ideas of Dugger's article Universal Homotopy Theories.

Suppose, we are given a nice model category $\mathcal{C}$, say left proper and cellular or combinatorial, so we have a good theory of localization. I am primarly thinking here of the category of presheaves of simplicial sets on some site with the projective model structure, where weak equivalences and fibrations are defined "pointwise".

Suppose furthermore, $S$ is a class of morphisms in $\mathcal{C}$ we can (left Bousfield) localize at [e.g. the class required for descent for hypercovers] and $T$ is an arbitrary set of morphisms in $\mathcal{C}$.

Now, let's consider a fibrant object $X$ in $\mathcal{C}[S^{-1}]$, i.e. some object which is $S$-local (and $\mathcal{C}$-fibrant), and take it's fibrant replacement $X^f$ in $\mathcal{C}[S^{-1}][T^{-1}]$.

Is it now reasonably to expect under some circumstances or even generally true that the map $X \to X^f$ is a weak equivalence in $\mathcal{C}[T^{-1}]$?

Answer 1

(Your question is basically about presentable (∞,1)-categories, so I will take the liberty of writing my answer in that language. Hopefully the translations to model category language will be straightforward.)

Inside $\mathcal{C}$ we have the full subcategories of $S$-local objects, $T$-local objects, and $(S \cup T)$-local objects, which are all reflexive subcategories; denote the associated localizations on $\mathcal{C}$ by $L_S$, $L_T$, $L_{S \cup T}$ respectively. Your question is whether for $X$ an $S$-local object, the map $X \to L_{S \cup T}X$ is a $T$-local equivalence, i.e., whether $L_T X \to L_T L_{S \cup T} X = L_{S \cup T} X$ is an equivalence in $\mathcal{C}$. In other words, the question is whether, if I start with an $S$-local object $X$, the localization $L_T X$ is still $S$-local.

In general this will not hold. For example, take $I$ to be the category $\ast \to \ast$ and let $\mathcal{C}$ be the diagram category $\mathrm{Fun}(I, \mathrm{Spaces})$. Write a typical object $X$ of $\mathcal{C}$ as $[X_1 \to X_2]$. There are sets $S$ and $T$ of morphisms such that the $S$-local objects are the ones for which the map $X_1 \to X_2$ is an equivalence and the $T$-local objects are the ones for which $X_2$ is a point (I think they're $S = \{[\emptyset \to \ast] \to [\ast \to \ast]\}$ and $T = \{[\emptyset \to \emptyset] \to [\emptyset \to \ast]\}$). Then $L_S[X_1 \to X_2] = [X_2 \to X_2]$ and $L_T[X_1 \to X_2] = [X_1 \to \ast]$. Clearly the $T$-localization of an $S$-local object need not be $S$-local.

One situation where I think the statement would hold arises from looking at models of a (finitary) essentially algebraic theory inside an ∞-topos. The idea is that the topos is a left exact localization of a category of presheaves of spaces, so that localization preserves the finite limits used to define the theory. However, that localization would need to be $T$, not $S$ as in your example, so I'm not sure whether this is the kind of example you had in mind.

Answer 2

Maybe to amplify on the structural aspects of the reply given by Reid Barton:

indeed, using Dugger's theorem the notion of Bousfield localization of combinatorial model categories precisely models the notion of localization of (oo,1)-category (of (oo,1)-presheaves), which is nothing but the notion of reflective (oo,1)-subcategory.

This is helpful, because it gives the somewhat ad hoc definition of Bousfield localization the more conceptual interpretation as a model for an adjunction

$ \mathbf{C} \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow} \mathbf{D}$

of (oo,1)-categories. From this perspective it is very plausible that given two such reflective embeddings $\mathbf{C}_1, \mathbf{C_2} \hookrightarrow \mathbf{D}$ there is no reason that the units of the corresponding adjunctions -- which are the localization morphisms -- have to satisfy any relation with each other, in general.