Questions tagged [stable-homotopy-category]

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7
votes
1answer
192 views

Is $[X, \_]$ a homology theory?

Let $X$ be a CW-spectrum. It is well-known that $[\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely,...
1
vote
0answers
120 views

Symmetric monoidal structure of the heart of $S^1$-spectra

How to give a symmetric monoidal structure of $SH^{S^1}(k)^{\heartsuit}$ (after $\mathbb{A}^1$-localization)? The standard answer is $$E_1\otimes E_2:=(E_1\wedge E_2)_{[0,0]}$$ but I don't see why ...
4
votes
0answers
103 views

Direct image and infinite suspension

I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
1
vote
1answer
163 views

Spectral sequence associated with a Postnikov tower (Solved by myself)

Suppose $E$ is an $S^1$-spectra of simplicial Nisnevich sheaves. For any $r\in\mathbb{Z}$, we have a distinguished triangle $$E_{\geq r+1}\longrightarrow E_{\geq r}\longrightarrow F_r\longrightarrow ...
5
votes
2answers
512 views

Why do we study complex orientable cohomology theories

It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology ...
3
votes
1answer
179 views

Infinite loop space of ring spectra: the cup product

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
11
votes
1answer
435 views

Equivalent definitions of Thom spectra

Background and notations: Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
3
votes
1answer
119 views

Basic question on the cobordism spectrum

I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...
7
votes
2answers
286 views

Model category structure on spectra

I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it. Let $S$ be a finite dimensional Noetherian scheme ...
6
votes
0answers
178 views

Compatible algebraic Spanier-Whitehead dual

Let me first ask an intuitive version of the question: Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...
11
votes
4answers
2k views

Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?

This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there). The answer ...
4
votes
1answer
328 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 $HA$,...
4
votes
0answers
80 views

Unstable and stable looping and delooping

I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...
1
vote
0answers
58 views

Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
2
votes
0answers
60 views

Does there exist a “Margolis-type” definition of equivariant cellular towers?

I am interested in cellular towers in the equivariant stable homotopy category $SH_G$ corresponding to a compact Lie group along with a complete universe for it. Note here that a cellular tower for ...
2
votes
1answer
183 views

Do Mackey (co)homology functors factor through derived categories? References with details?

Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...
3
votes
1answer
262 views

Basic questions on spectra

I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme. Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
10
votes
1answer
394 views

When is the Thom spectrum of a virtual vector bundle effective?

Remark: My question is valid in the classic setting of the stable homotopy category of spectra of CW-complexes. An answer on that setting will also be valid. Denote as $SH(X)$ Voevodsky's stable ...
11
votes
2answers
1k views

Connective spectra and infinite loop spaces

It seems to be standard that connective spectra are "the same" as infinite loop space. However, I do not understand the reason why the associated spectrum is connective. For me, an infinite loop ...
7
votes
0answers
182 views

Self-equivalences of the stable homotopy category

I have recently started approaching Stable Homotopy Theory and came up with what is probably a rather naive question to ask, though it looks like I can not find references on it around. Let $\...
7
votes
0answers
289 views

Is there a list of examples of orthogonal spectra?

Schwede's symmetric spectra book project provides point-set models of many important spectra as symmetric spectra, including (in §I.1) the sphere spectrum, Eilenberg-Mac Lane spectra, several Thom ...
3
votes
0answers
148 views

Brown Representability for Stable Homotopy Categories of Symmetric Spectra

Proposition 5.5 in $\mathbf{A}^1$-homotopy theory establishes the Brown's representability for the stable homotopy category $\mathcal{SH}_T(S)$, over a Noetherian scheme $S$, for a space $T$ of finite ...
2
votes
1answer
165 views

Symmetric spectra for simplicial sheaves

Let $C$ be a site. Localizing either the projective or injective model structure on simplicial presheaves at the Cech nerves of covers in $C$ gives a new model structure on simplicial presheaves which ...
11
votes
0answers
286 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
698 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
680 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
253 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
360 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
244 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
314 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 ...
9
votes
1answer
687 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. ...
20
votes
4answers
1k 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
264 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
326 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 (...
13
votes
1answer
462 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 ...
11
votes
0answers
497 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
394 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
314 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
391 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
685 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 ...
5
votes
1answer
463 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
181 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
133 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 ...
8
votes
1answer
357 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
478 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 ...
2
votes
2answers
282 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
456 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
401 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 ...
14
votes
2answers
954 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
439 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) ...