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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.
28
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3
answers
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What is the precise relationship between pyknoticity and cohesiveness?
Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement …
- 11.8k
22
votes
2
answers
6k
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References and resources for (learning) chromatic homotopy theory and related areas
What references and resources (e.g. video recorded lectures) are available for learning chromatic homotopy theory and related areas (such as formal geometry)?
19
votes
1
answer
1k
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$(\infty,2)$-categories: current applications and future prospects
Lately there has been a lot of progress on the foundations of $(\infty,2)$-categories (for example, all currently-known models for them were shown to be equivalent and finally we have a construction o …
- 11.8k
18
votes
1
answer
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Is the $\infty$-category of spectra “convenient”?
A 1991 paper of Lewis, titled “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$:
There is a symmetric monoidal smash …
- 11.8k
13
votes
0
answers
232
views
Isbell duality for simplicial sets
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv …
- 11.8k
12
votes
1
answer
5k
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How should one approach reading Higher Algebra by Lurie?
A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT sho …
11
votes
0
answers
872
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How far can Spectral Algebraic Geometry be developed over $\mathbb{E}_2$-rings (instead of $...
Jacob Lurie has extensively developed derived algebraic geometry in the setting of $\mathbb{E}_\infty$-ring spectra [SAG]. The resulting theory of Spectral Algebraic Geometry (SAG) gives (in particula …
- 11.8k
10
votes
1
answer
296
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Group completion of $\mathbb{E}_{\infty}$-monoids via tensor products
$\newcommand{\K}{\mathrm{K}}$The abelian group completion functor $\K_0\colon\mathsf{CMon}\to\mathsf{Ab}$ satisfies
$$
\K_0(A)
\cong
\mathbb{Z}\otimes_{\mathbb{N}}A,
$$
naturally in $A\in\mathrm{Obj}( …
- 11.8k
10
votes
1
answer
679
views
Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $ …
- 11.8k
10
votes
0
answers
316
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Near-ring spaces
$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1} …
- 11.8k
10
votes
1
answer
923
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What is the free symmetric monoidal $\infty$-category on one object?
It is well-known that the free symmetric monoidal category on one object is the category $\mathbb{F}$ of finite sets and bijections. This is supposed to be the categorification of the monoid of natura …
- 11.8k
8
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0
answers
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Classifying spaces of monoidal categories and deloopings
$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\bulle …
- 11.8k
8
votes
1
answer
634
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Grading ring spectra over the sphere spectrum
$\mathbb{Z}$-graded rings play an important role in algebra and algebraic geometry, so when moving to derived algebra and spectral algebraic geometry, it's natural to ask about ring spectra graded in …
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8
votes
1
answer
703
views
What is the homotopy category of the sphere spectrum?
Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?
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8
votes
1
answer
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Conservative cocompletion of categories of geometric shapes for homotopy theory
The recent paper
Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924
has shown that the conservative cocompletion of the simplex category $\Delta$ i …
- 11.8k