Questions tagged [bousfield-localization]

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Bousfield class of $TMF$ and $E(2)$

Let us work concretely at the prime 3. How does $TMF \wedge X \simeq E(2)$ for X a finite type 0 spectrum imply that $TMF$ and $E(2)$ have the same Bousfield class?
6 votes
1 answer
201 views

Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum

We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...
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8 votes
0 answers
319 views

Proving faithful flatness of a K-theoretic map without the moduli stack of formal groups

I'm in the process of writing an expository paper on complex K-theory and Snaith's theorem; the proof of Snaith's theorem that I'm following along (located at http://math.uchicago.edu/~amathew/snaith....
9 votes
2 answers
282 views

$p$-completeness of the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\infty} BK)$

Let $S$ be a finite $p$-group and $K$ a compact Lie group, in the paper A Segal conjecture for $p$-completed classifying spaces, it is said that the function spectrum $F(\Sigma^{\infty} BS, \Sigma^{\...
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3 votes
1 answer
162 views

$(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence

Let $\mathbb{Z}/p^{\infty}$ denote the Prufer group. By $p$-completion properties, it follows that $(B\mathbb Z/p^{\infty})^{\wedge}_p\simeq K(\mathbb{Z}^{\wedge}_p,2)\simeq(BS^1)^{\wedge}_p$. But, ...
  • 795
7 votes
1 answer
240 views

Interesting "epimorphisms" of $E_\infty$-ring spectra

$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ Suppose $i:A\to B$ is a map of $E_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod_B\to\Mod_A$ by ...
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6 votes
0 answers
211 views

When $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$?

When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^...
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1 vote
0 answers
246 views

Bousfield $p$-completion on spectra

Bousfield p-completion on spaces is a functor $(-)^{\wedge p}$ whose main property is that a map $f:X\rightarrow Y$ induces an isomorphism $f_{\ast}:H_\ast(X,\mathbb{F}_{p})\rightarrow H_\ast(Y,\...
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4 votes
0 answers
119 views

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 (...
1 vote
0 answers
75 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 ...
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3 votes
1 answer
206 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 votes
1 answer
232 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 ...
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5 votes
1 answer
189 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, ...
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5 votes
1 answer
211 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 ...
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5 votes
0 answers
109 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 ...
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7 votes
1 answer
281 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 ...
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15 votes
2 answers
840 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 votes
1 answer
250 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 ...
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6 votes
1 answer
811 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 $HA$,...
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15 votes
1 answer
580 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 ...
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5 votes
2 answers
355 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}...
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6 votes
1 answer
199 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....
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6 votes
0 answers
178 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-...
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18 votes
2 answers
845 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, ...
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8 votes
1 answer
408 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
1 answer
253 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 ...
5 votes
1 answer
235 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
1 answer
149 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
1 answer
428 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
0 answers
89 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 ...
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6 votes
1 answer
477 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 ...
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2 votes
1 answer
230 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 ...
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12 votes
2 answers
666 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{...
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