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Results tagged with homotopy-theory
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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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What facts in commutative algebra fail miserably for simplicial commutative rings, even up t...
Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear …
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23
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Derivators (in English)
Grothendieck, before he disappeared, was working on a manuscript called "Les Derivateurs", which detailed the theory of derivators. Prof. Cisinski has done work with them as he mentioned in this post …
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A combinatorial approximation functor sSet->qCat
Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap …
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Does derived algebraic geometry allow us to take quotients with reckless abandon?
So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes not schemes or algebraic spaces. The way I've seen this sort of thing r …
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Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?
Many topologists express a clear preference for working with CW complexes instead of simplicial sets.
One of the reasons is that the cellular chain complex of a CW complex is often easier to work w …
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Computing homotopies
Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to t …
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When does QCoh have 'enough perfect complexes'?
Let $X$ be a derived fpqc stack on the $\infty$-category of connective spectral affine schemes $\mathbf{Aff}^{\mathrm{cn}}=(\mathbf{Ring}^{\mathrm{cn}}_{E_\infty})^{\mathrm{op}}$, that is to say, a fu …
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The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete s...
Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying reflex …
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The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres
Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spher …
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What is the "universal problem" that motivates the definition of homotopy limits/colimits (a...
The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan extens …
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Is the simplicial completion of a localizer always a bousfield localization of the injective...
Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following axio …
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Example of a CW complex not homeomorphic to the realization of a simplicial set?
I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomor …
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Where to find the correct result in Higher Algebra, incorrect reference
I'm looking at the proof of Higher Algebra Proposition 6.1.6.27, and in the very first sentence of the proof, Lurie states:
The functor $(F\delta)_{\Sigma_n}$ is n-homogeneous by Proposition 6.1.5 …
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Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representabi...
Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form
$$\begin{matrix}
R&\to &T\\
\downarrow&{}^?\ne …
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Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf ...
Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the …
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