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 ...
- 25.2k
22
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 $n$ ...
- 3,784
10
votes
Accepted
Localizing spaces at stable homotopy equivalences
Because suspension spectra, the sphere spectrum and the Eilenberg-MacLane spectrum are all $(-1)$-connected, a map $f\colon X\to Y$ of spaces induces an isomorphism of stable homotopy groups if and ...
- 56.9k
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:
...
- 56.9k
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 ...
- 56.9k
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 ...
- 19.6k
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I have an unpublished note that proves Barwick's claim.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 ...
- 30.3k
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 ...
- 16.5k
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} ...
- 10.8k
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 ...
- 2,197
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 $\...
- 27.2k
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 ...
- 30.3k
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 ...
- 30.3k
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 ...
- 30.3k
4
votes
Does $\infty$-categorical localization commute with taking directed fibered products?
Here is a counter example in the general case:
Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence.
The lax-pullback is $\{id:1 \to 1\}$, and the ...
- 42.4k
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 ...
- 1,906
4
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)...
- 2,814
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 ...
- 37.8k
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. ...
- 63.9k
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.
...
- 63.9k
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 $B \simeq LA$ ...
- 9,263
2
votes
Transfer model structures, reflective subcategories and Bousfield localizations
Here is a counterexample to Hope 1.
Let $\mathsf M$ be the category of augmented dg algebras over a field of characteristic zero. Let $\mathsf N$ be the reflective subcategory of commutative algebras. ...
- 40.2k
2
votes
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 ...
- 9,111
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. ...
- 30.3k
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 \...
- 767
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 $...
- 63.9k
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