Questions tagged [bousfield-localization]

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Fibrant objects in $\mathbb{S}$-local model structure on $Top_*$

Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$. Equipping $Top_*$ with the Quillen model structure (...
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0answers
70 views

A characterization of local objects in model categories

Let $\mathcal{M}$ be a model category and $S$ a set of cofibrations between cofibrant objects. Then every $S$-local object has the right lifting property with respect to $S$. The converse does not ...
3
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1answer
147 views

Transfer model structures, reflective subcategories and Bousfield localizations

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 ...
2
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1answer
86 views

Definition of hypercover for simplicial presheaves and hypercovering in $\infty$-topos

By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of ...
4
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1answer
159 views

Homotopy fibre sequence and left Bousfield localization

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...
5
votes
1answer
185 views

Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves, how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...
5
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0answers
95 views

Cardinalities associated to the Bousfield lattice

By Ohkawa's theorem, the Bousfield lattice $B$ (of the $\infty$-category of spectra) is a small, complete lattice with $2^{\aleph_0} \leq |B| \leq 2^{2^{\aleph_0}}$ (the exact cardinality is an open ...
6
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1answer
240 views

Are these two notions of unstable localization suitably equivalent?

It seems to me that although homological localization (i.e. formally inverting $E$-homology equivalences for some $E$) is a reasonable thing to do to a spectrum, it's a pretty brutal thing to do to a ...
14
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2answers
651 views

Counter-example to the existence of left Bousfield localization of combinatorial model category

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists ...
3
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1answer
246 views

When localisation preserves isomorphy of homotopy groups

Let $E$ be a generalized cohomology theory. Let's agree that $E$ satisfies property $(*)$ if for any two finite CW complexes with isomorphic homotopy groups their $E$-localizations also have ...
4
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1answer
328 views

Intuition - difference between Moore spectrum and Eilenberg-Mac Lane spectrum

I know very little about algebraic topology, and more about $k$-linear stable $\infty$-categories (i.e. homological algebra). Given an abelian group $A$, there is the Eilenberg-Mac Lane spectrum $HA$,...
15
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1answer
487 views

Is a spectrum with trivial homology groups trivial?

If $X$ is a spectrum with trivial (integer-valued) homology groups, does it have to be weakly-equivalent to a point? This is easy to prove for connective spectrum, as a Hurewitz-type argument is then ...
5
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2answers
274 views

Limit of weak equivalences in a Bousfield localization

Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}...
6
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1answer
155 views

Derivator of localizations of spectra

Since my question is very specific let me introduce the context. I am trying to apply derivator theory to stable homotopy theory. Theorem 3 of the following preprint by Franke https://pdfs....
5
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0answers
144 views

The chromatic splitting conjecture and functoriality

Let $M$ be a finite spectrum, so that $L_{K(n)} M = M \wedge L_{K(n)} S$. Recall that (a weak version of) the chromatic splitting conjecture states that the chromatic attaching map $L_{n-1} M \to L_{n-...
18
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2answers
641 views

Detecting the Brown-Comenetz dualizing spectrum

The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 2$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, ...
8
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1answer
356 views

Left Bousfield localization without properness, what is known?

I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
5
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1answer
209 views

Factorization of Gabriel-Zisman localization construction?

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$. The localization ...
4
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1answer
150 views

Subcategories of the Verdier quotient?

Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$. Is there a bijective correspondence between ...
6
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1answer
137 views

Homotopy limit and Bousfield localization

Suppose $E$ is a spectrum and $p$ is a prime. We can then $(H\mathbb{Z}/p)_*$-localize to obtain $L_{H\mathbb{Z}/p}E$. Is it true that the natural map $L_{H\mathbb{Z}/p}E\rightarrow\text{holim}_n(L_{H\...
1
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1answer
413 views

Transfer of left Bousfield localization

My first question in this forum. Suppose we have a Quillen equivalence $L: M\leftrightarrow N: R$ between two model categories (say left proper cofibrantly generated cellular or locally presentable). ...
3
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0answers
86 views

Example of R-bad space

I have been looking around for examples of $R$-bad spaced in the sense of Bousfield and Kan. In their book "Homotopy limits, completions and localizations] they give several examples of such spaces ...
6
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1answer
384 views

Bousfield Localization and Quillen Equivalence

The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any ...
2
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1answer
195 views

How do you rigidify a Bousfield localization?

I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$. I thought there is a ...
12
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2answers
562 views

Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let $\mathbf{...