Questions tagged [stable-homotopy-category]
The stable-homotopy-category tag has no usage guidance.
22
questions with no upvoted or accepted answers
11
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381
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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 $...
11
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0
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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 ...
7
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264
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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 $S: \mathsf{C} \to \mathsf{C}$...
7
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210
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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{...
7
votes
0
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435
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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
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217
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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 ...
5
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120
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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 ...
5
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0
answers
144
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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 ...
4
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61
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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 ...
4
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190
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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 ...
4
votes
0
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100
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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 ...
4
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0
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368
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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....
3
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189
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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 ...
3
votes
0
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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
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139
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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}^{...
2
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0
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136
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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....
2
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0
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128
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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
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0
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234
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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 ...
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 ...
1
vote
0
answers
141
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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 ...
1
vote
0
answers
65
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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 ...
1
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0
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198
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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 ...