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Questions tagged [bousfield-localization]

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12
votes
1answer
334 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 ...
4
votes
1answer
224 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 ...
3
votes
1answer
145 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
votes
1answer
449 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
votes
2answers
229 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
votes
1answer
133 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
votes
0answers
122 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-...
19
votes
2answers
539 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
votes
1answer
282 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
votes
1answer
176 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 ...
3
votes
0answers
101 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
votes
1answer
127 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
vote
1answer
403 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
votes
0answers
79 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
votes
1answer
323 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
votes
1answer
181 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
votes
2answers
508 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{...