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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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Which properties of categories are preserved under homotopy equivalences?
This question arose from a discussion in the comment section of another MathOverflow question with Mike Shulman and Alec Rhea, which raised the following point:
Vague Question: Are adjunctions an app …
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The homotopy monoids of directed spheres
In directed homotopy theory, one replaces spheres by directed spheres and homotopy groups by homotopy monoids.
Is it known what are the first few homotopy monoids of directed spheres?
Do homotopy mon …
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What is the group completion of the groupoid of even finite sets and even permutations?
$\newcommand{\sgn}{\mathrm{sgn}}\newcommand{\defeq}{\overset{\mathrm{def}}{=}}$The Barratt–Priddy–Quillen–Segal theorem says that the $\mathbb{E}_\infty$-group completion of the groupoid of finite se …
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Looking for analogs of the rational, $p$-adic, and real numbers in homotopy theory via the p...
We can view the construction of the real or $p$-adic numbers as the end result of a series of constructions each of which adds more and more structure to the object at hand:
First, we start with the …
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How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $...
Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in particula …
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What do we know about the homotopy of $\mathrm{Q}S^0\wedge\mathrm{Q}S^0$?
The homotopy groups of $\mathrm{Q}S^0$ are the stable homotopy groups of spheres, of which we know the first 81 exactly, and the first 90 up to some uncertainties. A table is given in Wikipedia:
Do w …
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Restricting spectra to finite $n$-truncated/$n$-connected pointed spaces
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently I've noticed that the definitions of special $\Gamm …
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Weakening the excision condition for spectra
$\renewcommand{\S}{\mathcal{S}}\newcommand{\l}{\langle}\newcommand{\r}{\rangle}\newcommand{\op}{\mathsf{op}}\newcommand{\fin}{\mathrm{fin}}$Recently, I've noticed that the definitions of special $\Gam …
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$(-n-1)$-connected spectra vs. reduced excisive functors from $n$-truncated pointed spaces
It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of fi …
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Geometric models for the classifying spaces of the spin and string covers of the orthogonal,...
$\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\ …
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Can one define fields in stable homotopy theory via invertibility?
In Nilpotence in Stable Homotopy Theory II, Hopkins–Smith define a field spectrum to be a ring spectrum $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$. They then show that any field s …
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Homotopical properties of powersets of simplicial sets
Given a simplicial set $X_\bullet$, define its powerset simplicial set $\mathcal{P}_\bullet(X)$ as the composition
$$\Delta^\mathsf{op}\xrightarrow{X_\bullet}\mathsf{Set}\xrightarrow{\mathcal{P}}\math …
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Powersets of simplicial sets vs. powersets of topological spaces
Motivation. Recently I've been trying to understand how well- or ill-behaved are the many different powerset topologies one can put on the powerset of a topological space, and in particular I'm tryin …
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Is there a recognition principle for $\mathbb{E}_{\infty}$-spaces with zero?
A commutative monoid with zero is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal c …
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Applications of $\mathbb{Z}$-graded algebraic geometry to algebraic topology
There's a theory of algebraic geometry over $\mathbb{Z}_2$-graded commutative rings, often called "algebraic supergeometry" or the theory of superschemes. From what I understand, there's also a varian …
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