Questions tagged [stable-homotopy-category]
The stable-homotopy-category tag has no usage guidance.
64
questions
19
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2
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Why the sphere spectrum is more correct than $\mathbb{Z}$?
One may argue that $\mathbb{S}$ is more correct than $\mathbb{Z}$. Can anyone make it more explicitly? For example, what information will be lost if we work in $\mathbb{Z}$ instead of $\mathbb{S}$？
...
- 1,034
5
votes
0
answers
95
views
Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$
My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.
It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...
- 2,701
3
votes
0
answers
157
views
Compute the nearby cycles functor for the category of mixed motives
I am reading the survey of J. Ayoub, The motivic nearby cycles and the conservation conjecture (see here), in which he introduced the original version motivic nearby cycles (another note by Illusie is ...
- 436
2
votes
0
answers
119
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Geometric fixed points of induction spectrum
I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
- 717
2
votes
0
answers
119
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A basic computation with spectra
Let $\mathbb{E}=\big(E_n, \sigma_n\colon T\wedge E_n\to E_{n+1}\big)_{n\in\mathbb{N}}$ be a $T$-spectrum, either in the topological setting (with $T=S^1$) or in the algebraic setting (with $T=\mathbb{...
- 2,701
4
votes
0
answers
59
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Endomorphism in the rational stable $O(2)$-equivariant category of the universal space of the family of finite dihedral subgroups
Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a ...
- 717
8
votes
1
answer
418
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Inverting objects in a symmetric monoidal category
In Voevodsky’s ICM address:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf
In theorem 4.3 it is claimed that given a symmetric monoidal ...
- 103
6
votes
1
answer
334
views
Stable Adams operations
I have come across a paper by Adams, Harris and Switzer on the Hopf algebra of cooperations of real and complex K-theory. The Adams operations are stable in the $p$-local setting, however I have not ...
8
votes
1
answer
200
views
Bousfield's distributive lattice DL and non-ring spectra
Bousfield, in his paper "The Boolean algebra of spectra" (Comm Math Helv 54, 368–377 (1979), https://doi.org/10.1007/BF02566281), defined $\mathbf{DL}$, a sublattice of the Bousfield lattice,...
- 3,873
6
votes
1
answer
206
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Homotopy groups of $K(n)$-localization of the Brown-Peterson spectrum
We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we ...
- 717
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votes
0
answers
168
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Every spectrum is the homotopy colimit of shifted suspension spectra
Let $X$ be a spectrum. In various places, I have encountered the statement that
$$
X \simeq \text{hocolim}_n \Sigma^{\infty-n}X_n.
$$
I was wondering how this homotopy colimit is defined, and why we ...
- 173
7
votes
1
answer
360
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,...
- 795
1
vote
0
answers
135
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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 ...
- 814
4
votes
0
answers
166
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 ...
- 2,701
1
vote
1
answer
258
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
$$E_{\geq r+1}\longrightarrow E_{\geq r}\longrightarrow F_r\longrightarrow ...
- 814
8
votes
2
answers
874
views
Why do we study complex orientable cohomology theories
It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology ...
- 1,785
3
votes
1
answer
261
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Infinite loop space of ring spectra: the cup product
I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory.
Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
- 2,701
12
votes
1
answer
1k
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Equivalent definitions of Thom spectra
Background and notations:
Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
- 542
3
votes
1
answer
151
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Basic question on the cobordism spectrum
I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...
- 2,701
7
votes
2
answers
370
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Model category structure on spectra
I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional Noetherian scheme ...
- 2,701
6
votes
0
answers
206
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Compatible algebraic Spanier-Whitehead dual
Let me first ask an intuitive version of the question:
Let $Sp$ be the homotopy category of spectra. Let $E$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we ...
- 2,003
13
votes
4
answers
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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 ...
6
votes
1
answer
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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 $HA$,...
- 5,392
4
votes
0
answers
92
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 ...
- 2,701
1
vote
0
answers
60
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 ...
- 16.1k
2
votes
0
answers
71
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 ...
- 16.1k
2
votes
1
answer
221
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 ...
- 16.1k
3
votes
1
answer
284
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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 ...
- 2,701
10
votes
1
answer
477
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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 ...
- 2,701
11
votes
2
answers
2k
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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 ...
- 9,207
7
votes
0
answers
207
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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 $\mathcal{...
- 201
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votes
0
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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 ...
- 6,646
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votes
0
answers
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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 ...
- 333
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votes
1
answer
172
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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 ...
- 828
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votes
0
answers
357
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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 $...
- 423
6
votes
2
answers
874
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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 ...
- 2,701
11
votes
2
answers
711
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 ...
- 16.1k
3
votes
2
answers
275
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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 ...
- 16.1k
4
votes
1
answer
426
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{...
- 16.1k
3
votes
1
answer
272
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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 ...
- 2,701
8
votes
1
answer
340
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 ...
- 9,855
10
votes
1
answer
827
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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.
...
- 2,237
20
votes
4
answers
2k
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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 ...
- 2,237
4
votes
0
answers
342
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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....
- 7,377
6
votes
1
answer
371
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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 (...
- 3,281
14
votes
1
answer
603
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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 ...
- 489
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0
answers
580
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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 ...
- 9,855
8
votes
1
answer
434
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}_\...
- 12.1k
2
votes
1
answer
355
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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 ...
- 9,855
2
votes
1
answer
480
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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) $R$...
- 9,855