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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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Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets …
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84
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Do we still need model categories?
One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equ …
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35
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What do loop groups and von Neumann algebras have to do with elliptic cohomology?
Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:
Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of vec …
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A homotopy commutative diagram that cannot be strictified
By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to t …
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What is the homotopy theory of categories?
I've heard that Grothendieck, in his letter "Pursuing Stacks," wanted to find alternative models for the classical homotopy category of CW complexes and continuous maps (up to homotopy), and one of hi …
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20
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Acyclic models via model categories?
Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation $H …
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16
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A model category of spaces where strict commutative monoids are $E_\infty$-spaces
There are various strict monoidal model categories of spectra (e.g. symmetric spectra) where the honestly commutative monoid objects model the "coherently commutative" ring spectra (which might otherw …
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15
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Finite spectrum annihilated by multiplication by two
Let $X$ be a finite spectrum. Say that $X$ has characteristic two if multiplication by two on $X$ is nullhomotopic.
Does there exist a noncontractible finite spectrum of characteristic two?
(This …
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Strictly commutative elements of $E_\infty$-spaces
Let $X$ be an $E_\infty$-space (not necessarily grouplike). Let $x \in \pi_0 X$ be an element; say that $x$ is strictly commutative if there is a map of $E_\infty$-spaces $\mathbb{Z}_{\geq 0} \to X$ t …
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Homotopy groups of the orthogonal group
I'm interested in knowing what $n$-dimensional vector bundles on the $n$-sphere look like, or equivalently in determining $\pi_{n-1}(SO(n))$; here's a case that I haven't been able to solve.
Let $n …
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14
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Geometric interpretation of families in the stable homotopy groups of spheres
There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image …
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14
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Unobstructedness of braided deformations of symmetric monoidal categories in higher category...
Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue fiel …
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12
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$\mathcal{I}$-functors and infinite loop spaces
By the Barratt-Priddy-Quillen theorem, the space $B \Sigma_\infty^+$ is the infinite loop space $\Omega^\infty \Sigma^\infty S^0$. I'm curious about a "high-concept" reason that $B \Sigma_\infty^+$ (a …
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11
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Generating acyclic cofibrations for the various model structures in higher category theory
There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example:
The Joyal model structure on …
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11
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A homotopyish Landweber exact functor theorem
Let $M$ be a $\pi_*(MU)$-module. The Landweber exact functor theorem gives conditions for the functor that sends a space $X$ to $ MU(X) \otimes_{\pi_*(MU)} M$ to define a homology theory on spaces, wh …
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