Questions tagged [stable-homotopy-category]

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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 $...
Jack Davies's user avatar
11 votes
0 answers
629 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 ...
Jonathan Beardsley's user avatar
7 votes
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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 $S: \mathsf{C} \to \mathsf{C}$...
ಠ_ಠ's user avatar
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7 votes
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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{...
Marco Vergura's user avatar
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 ...
Arun Debray's user avatar
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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 ...
Prasit's user avatar
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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 ...
Tintin's user avatar
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5 votes
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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 ...
Jonathan Beardsley's user avatar
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 ...
N.B.'s user avatar
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4 votes
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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 ...
Tintin's user avatar
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4 votes
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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 ...
Tintin's user avatar
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4 votes
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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....
Ulrich Pennig's user avatar
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 ...
Alexey Do's user avatar
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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 ...
user24453's user avatar
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2 votes
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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}^{...
Alexey Do's user avatar
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2 votes
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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....
N.B.'s user avatar
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2 votes
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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{...
Tintin's user avatar
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2 votes
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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 ...
merle's user avatar
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2 votes
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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 ...
Mikhail Bondarko's user avatar
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 ...
Nanjun Yang's user avatar
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 ...
Mikhail Bondarko's user avatar
1 vote
0 answers
198 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 ...
Jonathan Beardsley's user avatar