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.
115
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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
$$\...
- 2,306
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An equation in *System Identification - Theory for the user* [closed]
I'm reading System Identification - Theory for the user and trapped in (2.66) below
I have no idea how the equation (2.66) developed. Thanks for any help!
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connective spectra vs. spectra
I am thinking about infinite loop spaces and spectra. The category of connective spectra is in fact equivalent to the category of infinite loop spaces.
Is it also the case that the category of spectra ...
- 4,122
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1
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Algebra with spectra: Vanishing in K(n) implies vanishing in a quotient of BP
I'm having a confusion about doing algebra with spectra. My question occured while reading p. 10 of Sanath Devalapurkar's Chromatic Homotopy Theory where we are in the course of proving the thick ...
- 4,122
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0
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Complex of sheaves, Eilenberg-MacLane spectra and hypercohomology
This question is about the relation between the category of spectra and the category of chain complexes of abelian groups. Specifically, I am trying to understand the examples from Deligne cohomology.
...
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Two Technical Details from "Categories and Cohomology Theories".
This has really bothered me for a while. I understand fully the big picture idea of what is happening in Segal's "Categories and Cohomology Theories". You take a Segal space $X$ and you ...
- 4,122
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1
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56
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Rational homotopy groups of spectra
It is claimed in a paper of Adams, Harris and Switzer that
$$\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 ...
- 7,687
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49
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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-...
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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 ...
CommunityBot
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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-...
CommunityBot
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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}$ ...
- 2,306
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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 ...
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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{...
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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\...
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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 *...
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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 ...
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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 ...
- 11k
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139
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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 ...
- 7,687
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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 (...
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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 ...
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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 $\...
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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 ...
- 437
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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\...
- 321
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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 ...
- 437
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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 ...
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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\...
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Homotopy groups of wedge sum
In the last chapter of his Concise course in algebraic topology, May states (without proof or reference) that for an arbitrary collection $(X_i)_{i\in I}$ of spectra the following hold:
$\pi_n(\prod_{...
- 2,554
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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 ...
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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$, ...
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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 (...
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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^...
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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 ...
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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 $\tilde{I}$, ...
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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 ...
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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) ...
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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)...
- 331k
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$\mathbf H R$ 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 ...
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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 $\...
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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 ...
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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 ...
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Spectral decomposition of an operator
Given the operator:
$$A=\begin{pmatrix}
i & 0 & - 4\\
0 & - 3i & 0\\
2 & 0 & - i
\end{pmatrix}$$
Now $\det(zI-A)=(z+3i)(z^2+9)=(z+3i)^2(z-3i)$,
I know a theorem that says that ...
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Showing that $\sigma_A (a) = \sigma_C (A).$
Let $A$ be a unital $C^*$-algebra and $C$ be a $C^*$-subalgebra containing the identity of $A,$ and let $a$ be a self-adjoint element in $C.$ Then $\sigma_A (a) = \sigma_C (a).$
It's quite clear that ...
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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, ...
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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 ...
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$\Omega$-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\\
\{ *\} ...
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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 ...
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141
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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 ...
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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 ...
- 141
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Commutativity of naive smash product
I am reading Switzer right now, and I came upon this line in the definition of naive smash product of spectra (p.256):
Some of the naive smash products are commutative ($E\wedge_{BC}F \simeq F\wedge_{...
- 11k
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How to analyze the differences between experimental and theoretical data.
I have around 5000 data points of theoretically computed and experimentally measured (via FTIR) K2 line positions(v (v", J''; v', J'). I divided the data into wn segments, so that I could analyze ...
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