# Questions tagged [stable-homotopy-category]

The stable-homotopy-category tag has no usage guidance.

53
questions

**7**

votes

**1**answer

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,...

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**0**answers

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**

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**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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**

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**1**answer

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

**1**answer

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**

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**2**answers

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**

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**0**answers

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

**4**answers

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

**1**answer

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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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**

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**0**answers

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**

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**0**answers

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**

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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

**1**answer

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 ...

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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 $...

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**2**answers

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**

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**2**answers

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 ...

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votes

**2**answers

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

**1**answer

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

**1**answer

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**

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**1**answer

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

**1**answer

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**

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**4**answers

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**

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**0**answers

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

**1**answer

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

**1**answer

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 ...

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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**

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**1**answer

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

**1**answer

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

**1**answer

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**

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**1**answer

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

**1**answer

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**

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**0**answers

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 ...

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**0**answers

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

**1**answer

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

**2**answers

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**

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**2**answers

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

**2**answers

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

**3**answers

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**

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**2**answers

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

**2**answers

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) ...