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10
votes
0answers
156 views

How does the HHR Norm functor interact with the cotensor over $G$-spaces?

Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces $...
6
votes
2answers
265 views

Smash product of spheres in $\mathbf{SH}$ and product in cohomology

I have two very concrete and simple question. Just in case I write downwards what led me into this. My questions: Let $\mathbf{SH}(X)$ be the stable homotopy category of Voevodsky. Denote $S^n$ the ...
11
votes
2answers
542 views

Do there exist “topologically significant” (and not “algebraic”) triangulated categories killed by the multiplication by $p$?

I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably ...
3
votes
2answers
239 views

Is the “inverse” (i.e., the “cohomological”) numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra “acceptable”? [closed]

I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider ...
4
votes
1answer
259 views

Does the (singular)cohomology of any acyclic spectrum vanish?

I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb{...
3
votes
1answer
169 views

Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
8
votes
1answer
276 views

Higher coherent multiplicative structures on S-algebras

In their book, Elmendorf, Kriz, May and Mandell describe a useful category of spectra, called S-modules, where S is the sphere spectrum. Ring objects in this category can be identified with spectra ...
8
votes
1answer
499 views

Morava $K(n)$'s are not $E_{\infty}$

I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right. ...
19
votes
4answers
827 views

Multiplicative Structures on Moore Spectra

The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
3
votes
0answers
220 views

matrix ring spectra

I am trying to understand matrix ring spectra. Apparently, I have two different definitions of those and I did not manage to show that they are equivalent - maybe they even are not in the general case....
6
votes
1answer
280 views

When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things: Homotopy groups stabilize after taking suspensions (...
12
votes
1answer
350 views

localizing subcategories of $HF_p$-local spectra

This entire question takes place in the $HF_p$-local category of $p$-local spectra, i.e. the essential image of $HF_p$-localization on the stable homotopy category. $HF_p$ itself is in there, and of ...
7
votes
0answers
322 views

Fields in Stable Homotopy Theory

It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these ...
8
votes
1answer
338 views

Stabilization of $\infty$-categories versus SW stabilization

Spanier-Whitehead stabilization provides a way to extend a category $\bf E$ to a bigger one $\mathcal{SW}_\Omega(\bf E)$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}_\...
2
votes
1answer
271 views

Counterexamples to Smallness of Harmonic Spectra

It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That ...
2
votes
1answer
309 views

Generators of Thick Subcategories

Suppose we are given a thick subcategory of the compact objects in the homotopy category of modules over a ring spectrum $R$. Are there conditions we can place on $R$, or on the category (compact) $R$...
10
votes
1answer
394 views

Smashing localizations in the category of spectra

Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization. The functor $L_E$ generally does not ...
4
votes
1answer
334 views

Is the stable homotopy category idempotent complete?

Is the stable homotopy category idempotent complete? I have not been able to prove it, and the proof for abelian groups seems to strongly rely on looking at elements. Thanks, Jon
1
vote
0answers
157 views

Well-Generated Localized Triangulated Categories

Suppose given a well-generated triangulated category with a compatible symmetric monoidal structure, $\mathcal{T}$ (in the sense of Neeman). Is it clear that the image of a localization functor will ...
4
votes
0answers
124 views

Bousfield Lattices for which Minimal Objects Coproduct to Sphere Object

Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield ...
6
votes
1answer
264 views

Is there an obvious reason why p-localization of spectra is a finite localization?

Is there an obvious reason why $p$-localization of spectra is a "finite" localization in the sense of Haynes Miller? In other words, is there an obvious reason why the localizing subcategory (of the ...
6
votes
2answers
401 views

Chromatic convergence of E(n)-localized homotopy categories

Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy ...
1
vote
2answers
239 views

Relationship of Bousfield Classes of Morava K-theories

Suppose we have $\langle K(n)\rangle$ and $\langle K(n-1) \rangle$ for some fixed prime $p$. do we know whether or not $\langle K(n) \rangle \geq \langle K(n-1) \rangle$ or $\langle K(n-1) \rangle \...
6
votes
2answers
396 views

Coreflective Subcategories of the Stable Homotopy Category

Here by stable homotopy category I mean the homotopy category of spectra, or maybe just some monogenic, Brown, algebraic, etc. stable homotopy category (in the language of Hovey, Palmieri and ...
3
votes
3answers
316 views

Bousfield Classes

This question has a few parts: 1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are ...
13
votes
2answers
687 views

Difficulties with the mod 2 Moore Spectrum

I have been informed that there is a reference out there which specifically details what goes wrong with the mod 2 Moore spectrum, i.e. why it is not $A_\infty$ or something? I do not know the details,...
6
votes
2answers
349 views

Does a finite suspension spectrum make a space finite?

Suppose that $X$ is a space whose suspension spectrum $\Sigma_+^\infty(X)$ is dualizable in the stable homotopy category. I believe this is equivalent to saying that $\Sigma_+^\infty(X)$ is (weakly) ...
31
votes
4answers
4k views

Definition of an E-infinity algebra

Can anyone give me a plain-and-simple definition of an E-infinity algebra without using the words "operad," "ring spectrum," or "stable homotopy"? Sorry, but I honestly couldn't find it using all on-...
15
votes
3answers
1k views

Is there an additive model of the stable homotopy category?

Is there a model category $C$ on an additive category such that its homotopy category $Ho(C)$ is the stable homotopy category of spectra and the additive structure on $Ho(C)$ is induced from that on $...
12
votes
2answers
573 views

Is there a constructive description of type in the p-local stable homotopy category?

The title pretty much sums it up - but let me give a little bit of background first. In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) ...