Questions tagged [stable-homotopy-category]

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1 answer
218 views

Symmetric-monoidal-associative smash product up to homotopy

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above. Recall that a sequential ...
7 votes
0 answers
264 views

Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
2 votes
2 answers
241 views

The complex $K$-theory of the Thom spectrum $MU$

The Atiyah-Hirzebruch spectral sequence is a strong computational tool that yields several interesting computation in (co)homology. I want to know whether $K_\ast(MU)$ and $K^\ast(MU)$ have been ...
2 votes
0 answers
139 views

Compatibility of different exchange structures $\operatorname{Ex}^*_{\#},\operatorname{Ex}^*_*,\operatorname{Ex}_{\# *}$

Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...
6 votes
0 answers
217 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 ...
22 votes
2 answers
3k views

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 $\mathbb{S}$? ...
20 votes
3 answers
2k views

Is there an additive model of the stable homotopy category?

$\DeclareMathOperator\Ho{Ho}$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 $\...
5 votes
0 answers
120 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 ...
3 votes
0 answers
189 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 ...
2 votes
0 answers
136 views

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....
2 votes
0 answers
128 views

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{...
4 votes
0 answers
61 views

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 ...
8 votes
1 answer
484 views

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 ...
6 votes
1 answer
435 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
216 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,...
6 votes
1 answer
248 views

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 ...
8 votes
2 answers
994 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 ...
2 votes
0 answers
234 views

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 ...
7 votes
1 answer
397 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
0 answers
141 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 ...
1 vote
1 answer
298 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 ...
4 votes
0 answers
190 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 ...
3 votes
1 answer
285 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 ...
13 votes
1 answer
2k 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
157 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
2 answers
414 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 ...
13 votes
4 answers
3k 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 ...
7 votes
1 answer
1k 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
0 answers
100 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
0 answers
65 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
0 answers
74 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
241 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
288 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
506 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 ...
12 votes
2 answers
2k 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
0 answers
210 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 $\mathcal{...
7 votes
0 answers
435 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
0 answers
171 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
177 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
0 answers
381 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
2 answers
945 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
2 answers
721 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
2 answers
284 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
468 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
277 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 ...
41 votes
4 answers
8k 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-...
8 votes
1 answer
344 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 ...
10 votes
1 answer
885 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. ...
21 votes
4 answers
2k 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 ...
4 votes
0 answers
368 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....