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 $\mathcal{C}[T^{-1}]$?

  • $\begingroup$ If this ever gets bumped to the front-page, I hope someone will retag with model-categories, higher-categories, and localization tags. I'm unwilling to bump it simply to re-tag. $\endgroup$ Sep 14 '11 at 16:48

(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.


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.


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