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Results tagged with homotopy-theory
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user 11546
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
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Whitehead Theorem for Harmonic Spectra
What are the chances that, for an arbitrary $p$-local harmonic spectrum $X$, if $K(n)\wedge X\simeq\ast$ for all $n$, then $X$ is contractible? This, I believe, holds for suspension spectra and finite …
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D(R) versus Ho(HR)?
Given an algebraic ring, how is its derived category related to the homotopy category of HR modules?
Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to …
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Essential maps of spectra which are null when localized at any prime
There are maps of spaces which are not null-homotopic, but when localized at any prime become null. I don't know explicit constructions of any, but an example is given in Section 6 of Chapter 25 of th …
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Localization at Infinite Wedges of K-theories or BP
This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges o …
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The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra
The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Rave …
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Geometric Interpretations of Homotopy Theoretical Constructions
In homotopy theory there are lots of nice constructions that seem designed to have some effect on the homotopy of a space, i.e. completing, localizing, and taking various homotopy (co)limits. It seems …
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Coherent MU_*-Modules
It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over $\Omega_\as …
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Thom isomorphism from the ABGHR perspective
In ABGHR Thom spectra are described in the following way: we start with a morphism of Kan complexes $X\to \mathbb{S}\text{-line}$, where $\mathbb{S}\text{-line}$ is an $\infty$-groupoid which is equiv …
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(Co)homology of a directed space with coefficients in a commutative monoid
This is essentially a reference request, or a request for an explanation of why this cannot be done in a useful or interesting way (i.e. an explanation of why no such reference exists!).
If I have a d …
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Invariant Ideals in Split Hopf Algebroids
Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group $\mathrm{ …
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Free Monoids in Closed Symmetric Monoidal Categories
There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.
Give some category $\mathcal{C}$ with the …
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Chromatic convergence of E(n)-localized homotopy categories
Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy cat …
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Monoidal Model Categories with Suspension Functor
This is basically just me trying to find out what such categories are called, and where they are written about. If I think of some model category of spectra being a "stabilization" of some model cate …
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Right Notion of Localizing Subcategory in Quasicategory
Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and re …
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Thom Spectra and Hopf-Galois Extensions of Ring Spectra
So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it …
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