26 votes
Accepted

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

A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the ...
21 votes
Accepted

Is a spectrum with trivial homology groups trivial?

If $K(n)$ is the $n$-th Morava K-theory for $n>0$, then $K(n)\otimes H\mathbb{Z}=0$ because, via the 2 complex orientations of $K(n)\otimes H\mathbb{Z}$, there are two formal groups over the ring $\...
12 votes
Accepted

Detecting the Brown-Comenetz dualizing spectrum

Your question appears to be equivalent to the 'dichotomy conjecture' of Hovey, which I believe is still open. First, note that any finite spectrum has a type, and all finite spectrum of type $n$ ...
  • 3,595
10 votes
Accepted

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

See Lemma 2.3 of the following paper, and the surrounding discussion: ...
10 votes

Detecting the Brown-Comenetz dualizing spectrum

I don't think that the answer is known. However, here are some comments. I will work everywhere with $p$-local spectra, for some fixed prime $p$, and write $I$ for the $p$-local Brown-Comenetz ...
8 votes

Limit of weak equivalences in a Bousfield localization

In the language of $\infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This ...
  • 18.9k
7 votes
Accepted

Left Bousfield localization without properness, what is known?

I have an unpublished note that proves Barwick's claim. Aspects of this story have appeared in some papers of mine with Michael Batanin, including one we published in the proceedings of the 2015 CRM ...
  • 22.6k
7 votes
Accepted

Bousfield Localization and Quillen Equivalence

For (1)-(2), look at work of Carles Casacuberta. He has lots of good examples. His paper with Chorny on the orthogonal subcategory problem has an example for your (2), on the last page. This paper of ...
  • 22.6k
7 votes
Accepted

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

I'm not sure I can answer to the general question, but I can explain why the Moore spectrum $\mathbb{S}/p$ shows up in the discussion of Bousfield localizations. This is just the cofiber of ...
  • 15.7k
7 votes
Accepted

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

Think of the Prüfer group as $\mathbb{Q}/\mathbb{Z}$ and of $S^1$ as $\mathbb{R}/\mathbb{Z}$. Here $\mathbb{Q}$ is discrete, $\mathbb{R}$ has its usual topology and is contracible. The map $\mathbb{Q} ...
  • 6,164
6 votes

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

Tim's argument is correct, and here's a different way to see this. To say that $\Sigma^{\infty} BS$ is a finite $p$-group has trivial rationalization and is $p$-local is the same as to observe that it ...
6 votes
Accepted

Homotopy limit and Bousfield localization

Yes, $H\mathbb{F}_p$-localizations are $p$-complete. We have the cofiber sequence $$ \Sigma^{-1}\mathbb{S}/p^\infty \xrightarrow{j} \mathbb{S}\to \mathbb{S}[p^{-1}] \to \mathbb{S}/p^\infty $$ where $\...
  • 25.8k
5 votes

Simple question: different definitions of Bousfield localization

There are many equivalent ways of defining the local equivalences. Let $\mathcal{M}$ be a simplicial model category with a cofibrant replacement functor $Q : \mathcal{M} \to \mathcal{M}$. Then, for a ...
  • 13.4k
5 votes

Homotopy fibre sequence and left Bousfield localization

The idea you're looking for is called "fibrewise localization". It's defined in Dror Farjoun's book "Cellular Spaces, Null Spaces, and Homotopy Localization", and also in Hirschhorn's book (since you ...
  • 22.6k
5 votes

When localisation preserves isomorphy of homotopy groups

The question is very unlikely to have a satisfactory answer in the level of generality asked. Let $X$ and $Y$ be the two spaces with isomorphic homotopy groups, and let $LX$ and $LY$ denote their ...
  • 22.6k
5 votes

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

While preparing the paper Left Bousfield localization without left properness, I learned another example, due to Voevodsky. It's example 3.48 in his paper Simplicial radditive functors. This is an ...
  • 22.6k
4 votes
Accepted

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

The condition that $U_n$ being a coproduct of representables is not essential for the definition of hypercover of simplicial presheaves. In fact, in Jardine's Local Homotopy Theory, a hypercover is ...
  • 1,624
4 votes

How do you rigidify a Bousfield localization?

There's an underived version of the story that might be worth working through first or instead. Namely, for a category $C$, there's a natural correspondence between reflective subcategories of $C$ and ...
4 votes
Accepted

Simple question: different definitions of Bousfield localization

Yes, they are the same. In order to prove it, we should show that they have the same new weak equivalences (this is enough, because both have the same cofibrations). Have a look at Barwick's paper On ...
  • 22.6k
3 votes

Subcategories of the Verdier quotient?

Yes, this is true and works as you expect. See Proposition 2.3.1 (pages 125-127) of Verdier's thesis: Jean-Louis Verdier -- Des catégories dérivées des catégories abéliennes (Astérisque No. 239, 1996)...
3 votes

Transfer of left Bousfield localization

Yes. This is Theorem 3.3.20 in Hirschhorn's book, if by $T$ you mean the left derived maps of $L(S)$, e.g. cofibrant replacements of the maps $L(S)$.
  • 22.6k
3 votes
Accepted

Limit of weak equivalences in a Bousfield localization

No. For a counterexample to your claim, consider the model category M of simplicial presheaves on a small site S equipped with the projective model structure. Its fibrant objects are presheaves of ...
3 votes
Accepted

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

Here is an argument (not as clean as Piotr's below). Use with caution; it's possible that I've made a mistake. We don't use any properties of $\Sigma^\infty BK$ -- this could be an arbitrary spectrum. ...
  • 51k
2 votes

Are these two notions of unstable localization suitably equivalent?

Let $E \neq 0$ be a spectrum. Here is a classification of the strong $E$-local equivalences, a proof that the localization with respect to them exists, and an analysis of some special cases. ...
  • 51k
2 votes
Accepted

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

If $A$ is an $E_\infty$ ring spectrum and $i : A \to B$ is any map of $A_\infty = E_1$ ring spectra such that the multiplication $\mu : B \wedge_A B^{op} \to B$ is an equivalence, then $B \simeq LA$ ...
  • 7,411
1 vote
Accepted

Transfer model structures, reflective subcategories and Bousfield localizations

So far, the best results that I managed to find in the direction of my questions appear both in some paper of Boris Chorny and his collaborators. 1st. Theorem, A non-functorial Bousfield-Friedlander ...
1 vote
Accepted

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

First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "homotopy pullback." In your question, you are missing a step. ...
  • 22.6k
1 vote

Derivator of localizations of spectra

I think I found a proof which uses only derivator theory. For sake of simplicity we deal the case $P=1$, but the proof clearly will be the same for any $P$. Associated to the coherent morphism $f \...
  • 705
1 vote

Factorization of Gabriel-Zisman localization construction?

I agree with Tyler that it's not clear what the universal property of, say $S^l C$ is supposed to be. Does a functor $G: S^l C \to D$ correspond to (i) a functor $F: C \to D$ with the property that $...
  • 51k

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