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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 is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic geometr …
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41
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Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwilli...
Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be i …
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35
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2
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What is the relationship between connective and nonconnective derived algebraic geometry?
"Derived algebraic geometry" usually means the study of geometry locally modeled on "$Spec R$" where $R$ is a connective $E_\infty$ ring spectrum (perhaps with further restrictions). Why "connective", …
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31
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Why is the motivic category defined over the site of smooth schemes only?
Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a cas …
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27
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Spectral sequences as deformation theory
I believe that running the spectral sequence of a filtered complex / spectrum $ \cdots \to F_n \to F_{n+1} \to \cdots$ can be viewed as doing deformation theory in some very primitive "derived algebra …
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26
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Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, …
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25
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What are Koszul dualities?
I am bewildered by the number of things I've heard referred to as "Koszul duality", and I would like to sort it out. At various different times, I believe I've seen any of the following phenomena refe …
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23
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What are some toy models for the stable homotopy groups of spheres?
The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" for the r …
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23
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What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?
Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?
Recall that a stable $\infty$- …
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22
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What's with equivariant homotopy theory over a compact Lie group?
For some reason -- I'm not quite sure why -- I've developed the impression that I'm supposed to "tiptoe" around equivariant homotopy theory over groups that are not finite. Should I?
Let me explain. …
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22
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Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy th …
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21
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3
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Why is Kan's $Ex^\infty$ functor useful?
I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful …
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20
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What is the group completion of finite sets with respect to cartesian product?
Let $\Sigma_+$ be the groupoid of finite pointed sets. The Barratt-Priddy-Quillen theorem tells us that the group completion of $\Sigma_+$ with respect to the symmetric monoidal structure given by wed …
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19
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Is every limit-closed, accessibly-embedded full subcategory of a presentable $\infty$-catego...
Let $C$ be a presentable $\infty$-category and let $D\subseteq C$ be a full subcategory closed under limits and sufficiently-filtered colimits. If $D$ is known to be accessible, then by the adjoint fu …
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Example of an unstable map between finite complexes which is the identity on homotopy but no...
Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it's not known if they exist between finite complexes (Freyd's Generating Hypothesis). Unstably, it's easy to find maps which …
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