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Questions tagged [spectra]

Use this tag for questions concerning the notion of spectra in algebraic topology. Spectra are natural objects which arises when dealing with stable properties. For spectra of linear operators, please use the tag (spectral-theory) instead.

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Is there a "nice" description of Bousfield localisation at $H \mathbb{Z}$?

Let $\newcommand{\Sp}{\mathrm{Sp}}\Sp$ denote the $\infty$-category of spectra and $\newcommand{\Z}{\mathbb{Z}} H\Z$ the Eilenberg-Mac Lane spectrum of the integers. Given any spectrum $X \in \Sp$, I ...
Ben Steffan's user avatar
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Adjoint triplet induced by exact functor of stable categories

For $\mathscr{A}$ a small stable $\infty$-category, we can consider the following diagram: where $\mathcal{Y}_\mathscr{A}$ denotes the ordinary Yoneda embedding $\mathscr{A} \to \mathcal{P}(\mathscr{...
h3fr43nd's user avatar
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Collapse of the AHSS for $E^*(\mathbb{C}\mathrm{P}^n)$ for $E$ a complex-oriented ring spectrum [duplicate]

Denote by $\newcommand{\CP}{\mathbb{C}\mathrm{P}}i_{n, m}\colon \CP^n \to \CP^m$, $n \leq m \leq \infty$ the canonical inclusion. Let $E$ be a homotopy commutative ring spectrum and $t \in \tilde{E}^2(...
Ben Steffan's user avatar
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3 votes
1 answer
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Stable categories are tensored over spectra

It's a folklore result by Lurie that for a (presentable) stable $\infty$-category $\mathscr{C}$ the mapping space functor $\mathsf{Map}:\mathscr{C}^{\mathrm{op}} \times \mathscr{C} \to \mathsf{An}$ ...
Qi Zhu's user avatar
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2 votes
1 answer
150 views

Forgetful functor from derived category $D(\mathbb{Z})$ to Spectra

How is concretely defined the (canonical?) forgetful functor from $D(\mathbb{Z})$, the derived category of the ring of integers, to the catetegory of spectra $\text{Sp}$? (here is refered to such map. ...
user267839's user avatar
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2 votes
1 answer
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Defining (co)homology groups of sequential spectrum

I just started learning about spectra. I know only the following definition of the sequential spectrum. A sequential spectrum $E$ is a sequence of spaces $E_n$ with basepoint, provided with maps $$\...
Haldot's user avatar
  • 820
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An equation in *System Identification - Theory for the user* [closed]

I'm reading System Identification - Theory for the user and trapped in (2.66) below I have no idea how the equation (2.66) developed. Thanks for any help!
John Ham's user avatar
1 vote
1 answer
77 views

connective spectra vs. spectra

I am thinking about infinite loop spaces and spectra. The category of connective spectra is in fact equivalent to the category of infinite loop spaces. Is it also the case that the category of spectra ...
user avatar
2 votes
1 answer
82 views

Algebra with spectra: Vanishing in K(n) implies vanishing in a quotient of BP

I'm having a confusion about doing algebra with spectra. My question occured while reading p. 10 of Sanath Devalapurkar's Chromatic Homotopy Theory where we are in the course of proving the thick ...
Qi Zhu's user avatar
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5 votes
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Complex of sheaves, Eilenberg-MacLane spectra and hypercohomology

This question is about the relation between the category of spectra and the category of chain complexes of abelian groups. Specifically, I am trying to understand the examples from Deligne cohomology. ...
timaeus's user avatar
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2 votes
1 answer
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Two Technical Details from "Categories and Cohomology Theories".

This has really bothered me for a while. I understand fully the big picture idea of what is happening in Segal's "Categories and Cohomology Theories". You take a Segal space $X$ and you ...
Johnathon Taylor's user avatar
3 votes
1 answer
67 views

Rational homotopy groups of spectra

It is claimed in a paper of Adams, Harris and Switzer that $$\pi_*E \otimes \pi_*F \otimes \mathbb{Q} \to E_*F \otimes \mathbb{Q}$$ is an isomorphism. This map is constructed by taking the map $\pi_*E ...
categorically_stupid's user avatar
2 votes
0 answers
67 views

p-completion preserves cofiber sequences

Suppose I have a cofiber sequence $X \to Y \to Z$ of spectra in the stable homotopy category. I want to show that there is still a cofiber sequence $X^\wedge_p \to Y^\wedge_p \to Z^\wedge_p$ after p-...
categorically_stupid's user avatar
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26 views

Sequential spectra formed from applications of B

I am interested in the Ω-spectrum Xᵢ, Xᵢ ≅ $ΩX_{i+1}$, where $Xᵢ := BⁿX$ for a CW-complex $X$. This construction is left adjoint, right? It seemed like it wouldn't be very hard to construct the smash ...
user avatar
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Limitations of Ω-Spectra

Right now I am interested in the "stabilization" endofunctor of the category of ∞-groupoids sending an object $X$ to $\text{colim } Ωⁿ Σⁿ X$. This colimit is related to $∃Y:X≅ΩY$. In Ω-...
user avatar
1 vote
1 answer
65 views

Function of spectra and cofinal subspectra

Let $E$ and $F$ be CW-spectra and $\tilde{F}$ be cofinal. Let $f \colon E \to F$ be a function of spectra. I am trying to prove: There is $\tilde{E} \subseteq E$ cofinal such that $f$ maps $\tilde{E}$ ...
Candyblock's user avatar
1 vote
1 answer
37 views

Preimage of cellcomplex is cellcomplex

I am trying to prove the following from Adams blue book (Lemma 2.6.(i)): Let $E$ and $F$ be CW-spectra, and $\tilde{F} \subseteq F$ be cofinal. Then there is a cofinal subspectrum $\tilde{E} \subseteq ...
Candyblock's user avatar
2 votes
3 answers
253 views

Homotopy groups of wedge sums of spectra

This question came up when I was trying to understand Lemma 2.2.9 in Barnes & Rotzheim, which states that for any set of (sequential) spectra $X_i$, the natural map $$\bigoplus_i\pi_n(X_i)\to\pi_n\...
Tipping Octopus's user avatar
1 vote
1 answer
107 views

Is the full subcategory of $p$-local finite spectra a thick subcategory of all finite spectra?

I am trying to understand Balmer's classification of the spectrum of the category $\mathsf{Sp}^\text{fin}$ of finite spectra. The inclusion $\mathsf{Sp}^\text{fin}_{(p)} \subseteq \mathsf{Sp}^\text{...
Jonas Linssen's user avatar
5 votes
0 answers
71 views

Nullhomotopicity of $\mathbb{S}/p \to^p \mathbb{S}/p$ for $p=2$ and $p \neq 2$?

For a given spectra $X$ we have $X/p$ defined as the cofiber $X \to^p X$ where the map is basically defined via defining it on the sphere spectrum $\mathbb{S}$! To define $\cdot p$ on the sphere ...
user135743's user avatar
2 votes
1 answer
142 views

Why does a stable category admitting finite limits, filtered colimits and $\Sigma$ admit pushouts?

I'm trying to understand how to compute pushout in Spectra. The reason it should satisfy it is because a stable category admitting finite limits, filtered colimits and $\Sigma$ (i.e pushout of $X \to *...
user135743's user avatar
2 votes
1 answer
122 views

When is a $p$-local spectrum zero?

I am currently reading Lurie's notes on chromatic homotopy theory and fail to see the following remark in lecture 26: Remark 5. Let $X$ be a finite $p$-local spectrum. Then $H_\ast(X,\Bbb F_p) \simeq ...
Jonas Linssen's user avatar
2 votes
2 answers
164 views

If Spectra are analogous to chain complexes, why do we have both $KG(\mathbb{Z}), \mathbb{S}$

I've heard that $Sp$ is analogous to the derived category $Der(Ch\mathbb{Z})$ (I will thus refer to those two categories as the left and right side respectively below). Namely, every spectra is a ...
user135743's user avatar
0 votes
1 answer
59 views

Induction preserves weak equivalences

Let $G$ be a finite group and $H \leq G$ be a subgroup. There is an induction functor $G \ltimes_H - : \mathbf{Sp}^H \to \mathbf{Sp}^G$ from the category of $H$-spectra to the category of $G$-spectra (...
Qi Zhu's user avatar
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1 vote
1 answer
131 views

Stable homotopy groups commute with inverse limit

Suppose we have a family of spectra $(E_i)_{i \in I}$ such that the inverse limit $\lim_i E_i$ does exist in the stable homotopy category (i.e. $\lim_i E_i$ is the limit in $\mathrm{SHC}$, the stable ...
Candyblock's user avatar
2 votes
1 answer
266 views

Why is the Steenrod algebra isomorphic to the cohomology of the Eilenberg MacLane spaces?

I will stick to $p=2$. I define the Steenrod algebra to be the algebra of (topological) stable cohomology operations modulo 2. I've found in the literature the identification of the Steenrod algebra $\...
groupoid's user avatar
  • 362
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0 answers
34 views

Suspension functor of pro-spaces to pro-spectra preserves weak equivalences

Suppose we have the category $\mathrm{sSet}$ of simplicial sets and $\mathrm{sSpectra}$ of simplicial spectra. Then we have the functor $\Sigma^{\infty} \colon \mathrm{sSet} \to \mathrm{sSpectra}$, $X ...
Candyblock's user avatar
1 vote
0 answers
112 views

Cohomology ring as a coefficient ring

For a space $X$ and a spectrum $E$, we define $X\wedge E$ as the spectrum with $n$th space $X\wedge E_n$ and obvious connecting maps. I wonder if the following identification holds in general: $$H^n(X\...
timaeus's user avatar
  • 321
1 vote
0 answers
65 views

Spectra and cohomology

I am trying to understand spectra and their relation to cohomology theory. I have read that there is an essentially surjective functor $\mathrm{coTheo} \to \mathrm{Spec}$ from the category of ...
Candyblock's user avatar
4 votes
1 answer
161 views

If $M$ is an $R$-module, how can I show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum

Let $R$ be a commutative ring with identity, and let $M$ be an $R$-module. I want to show that the Eilenberg-Maclane spectrum $HM$ is an $HR$-module spectrum. Specifically, I want to know how to ...
Austin Maison's user avatar
3 votes
2 answers
76 views

Why aren't spectra functions adequate to be the maps in the spectra category?

In his book, "Algebraic Topology - Homotopy and Homology", Switzer define (Definition 8.9) spectra functions as $f:E\to F $ is a function if is a collection of cellular maps, $\{f_n:n\in\...
Bianca Oliveira's user avatar
3 votes
1 answer
143 views

Proving some properties of the localization functor in the stable homotopy category.

I am trying to understand the paper named " Localization with respect to Certain Periodic Homology Theories" Here is the part of it I am trying to understand the proof of proposition 1.5 in ...
weird's user avatar
  • 29
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0 answers
85 views

Does the generalized homology represented by the sphere spectrum give the stable homotopy groups on suspension spectra of pointed spaces?

If $\mathbb{S}$ is the sphere spectrum, I would like to show that the generalized homology theory represented by $\mathbb{S}$, when evaluated on the suspension spectrum $\Sigma^\infty X$ of a pointed (...
I.A.S. Tambe's user avatar
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1 vote
1 answer
131 views

Cohomology operations are group homomorphisms

Let our spaces have the homotopy type of a CW complex, and let $E^*,F^*$ be two cohomology theories. A (degree $n$, stable) cohomology operation is a map $\Phi: E^q(X) \to F^{q+n}(X)$ for each $q$, ...
Descartes Before the Horse's user avatar
3 votes
1 answer
90 views

Notations for Whitehead tower in Anderson duality

In appendix B of Hopkins and Singer's paper, Lemma B.15., the authors claimed that we can deduce the following isomorphisms $$ [X, \Sigma^n \tilde{I}]\rightarrow [X \langle n-1, \infty\rangle , \Sigma^...
timaeus's user avatar
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2 votes
2 answers
110 views

The monoidal structure on the fundamental groupoids of spectrum

I am trying to understand Anderson duality and Picard categories from appendix B of Hopkins and Singer's paper, and I somehow get stuck on Example B.7 (Page 87). For a spectrum $E$, they consider each ...
timaeus's user avatar
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2 votes
0 answers
215 views

Confusions about function spectrum in Anderson duality

In appendix B of this paper https://arxiv.org/abs/math/0211216, Hopkins and Singer defined the Anderson dual $\tilde{I}(E)$ of a spectrum $E$ as the function spectrum of maps from $E$ to $\tilde{I}$, ...
timaeus's user avatar
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1 vote
1 answer
123 views

Why is $X \mapsto hom(\pi_*^{st}X, \mathbb{Q})$ the same as ordinary rational cohomology?

I am trying to understand the notion of Anderson duality from appendix B of this paper https://arxiv.org/abs/math/0211216 by Hopkins and Singer. But I somehow get stuck at the very first steps. I am a ...
timaeus's user avatar
  • 321
3 votes
2 answers
304 views

Loop-Suspension adjunction unit is stable equivalence

Let $X$ be a sequential spectrum, then the unit of the $(\Sigma,\Omega)$-adjunction yields a map $\eta : X \to \Omega \Sigma X$. The authors of the book Foundations of Stable Homotopy Theory (p. 48) ...
Qi Zhu's user avatar
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0 votes
1 answer
69 views

Definition of a map of (pre)spectra in HoTT

I'm trying to pick up some homotopy theory by reading various HoTT sources (I find it much easier than reading classical homotopy theory books). In Floris Van Doorn's PhD thesis he defines (pre)...
ಠ_ಠ's user avatar
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1 vote
0 answers
75 views

$\mathbf H R$ is a ring spectrum if $R$ is a ring

The first example of a ring spectrum is probably the Eilenberg-McLane spectrum of a ring $R$. But how is the multiplication $\mu: \mathbf H R \wedge \mathbf H R \to \mathbf H R$ defined? Probably this ...
arnett's user avatar
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5 votes
2 answers
289 views

What is the essential image of the suspension spectrum functor $\Sigma^\infty$?

Let $\mathsf{hCW}$ denote the homotopy category of CW-complexes and $\mathsf{hCWSpec}$ the homotopy category of CW-spectra (ie. families of CW-complexes $(X_i)_{i\in\mathbb{Z}}$ with connection maps $\...
Jonas Linssen's user avatar
2 votes
1 answer
105 views

Excision in the stable homotopy category

Is there a way to make precise the statements (if it holds at all) that excision holds in the stable homotopy category? I am a beginner in this kind of things, and the stable homotopy category for me ...
arnett's user avatar
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0 answers
140 views

Magnitude Spectrum from real 2D Fourier Coefficients

I know how to calculate the coefficients for a 2D Fourier Series as shown: 2D real Fourier Series I calculated the Coefficients A[m][n], B[m][n], C[m][n] and D[m][n]. Now I want to create a magnitude ...
dh21's user avatar
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3 votes
1 answer
96 views

Notation clarification in stable homotopy theory

In this paper which surveys the Goodwillie Calculus, the following notation is used in the introduction: $$\pi_*(P_1 I(X)) \cong \pi_*^s(X)$$ where $X$ is a based space, $I$ is the identity functor, ...
Matt's user avatar
  • 3,326
2 votes
1 answer
156 views

Discrete Spectra

What is a "discrete spectra" in context of homotopy theory/ derived category theory? It is for example mentioned here. Although it looks quite "googleable" I found nowhere a ...
user267839's user avatar
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0 votes
1 answer
71 views

$\Omega$-spectrum from homotopy-cartesian squares?

Let $\{X_{n}\}_{n\geq0}$ be a sequence of (pointed) spaces such that for all $n \geq 0$, we have homotopy-cartesian squares $\require{AMScd}$ \begin{CD} X_{n} @>>> \{* \}\\ @VVV @VVV\\ \{ *\} ...
Sunny Sood's user avatar
8 votes
1 answer
152 views

Visualising Spectra?

I have recently started to learn about Spectra. To state the definition (that I have learned), a spectrum $X = \{X_{n} \}_{n \geq0}$ is a sequence of based spaces $X_{n}$, with basepoint preserving ...
Sunny Sood's user avatar
1 vote
0 answers
49 views

Checking $X \to S[J^{-1}] \wedge X$ is an $S[J^{-1}]$ equivalence.

This is surely a trivial question, since all sources seem to relegate it to an exercise. Here's my attempt at a proof. Is this the idea? Please excuse the wordiness of the exposition, it's from my own ...
Jordan Levin's user avatar
4 votes
0 answers
126 views

Spectra smashing with $\mathbb{S}^1$ is equivalent to shifting

I'm struggling with understanding the proof that the suspension isomorphism in the homotopy category of topological spectra is equivalent to smashing with the sphere $\mathbb{S}^1$. The proof I'm ...
Nissokam's user avatar
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