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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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An adjunction between monads on $\mathcal{C}$ and presentable categories under $\mathcal{C}$
Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!).
I'm looking for a reference for the fol …
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245
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Uniqueness of the $(2,2)$-category theory of $(\infty,1)$-categories?
The question, as in the title, may be very simply stated as follows:
Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto eq …
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12
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403
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The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from tha...
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the h …
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6
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1
answer
341
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Compact objects in the $\infty$-category presented by a simplicial model category
Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-categori …
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5
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335
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A compendium of weak factorization systems on $sSet$
A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying
Every morphism $f:x \to y \in \mathcal{C}$ can be factored (n …
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219
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Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in...
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element …
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12
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1
answer
853
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The (fiber of the) cofiber of the fiber of a map of spaces
Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point se …
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210
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Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data
Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtr …
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75
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Bounding the dimension of the euclidean space in which any $n$-manifold embeds "$k$-uniquely...
(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the funct …
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A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)
Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the …
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14
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2
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774
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Interpretation of the cohomology of compact lie groups and their classifying spaces in DAG?
I'll be using homological grading throughout this question.
Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:
$$H^{\bullet}( …
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3
votes
1
answer
149
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A "non-abelian excision" statement for mapping out of a space
Let $U \subset A \subset X$ be spaces (in the sense of homotopy theory).
For every pointed space $Y$ restriction maps induce the following canonical map between mapping spaces:
$$fiber(Map(X,Y) \to …
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344
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Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?
I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:
What's the neat abstract framework for obstruction theory for non-abelian …
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1
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950
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Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of ...
There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework.
To state them let $G$ be a group acting on a connected (1-truncat …
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Generalities on sheaves - Where can I find the technology that can make this "proof" of Atiy...
Fix $R$ an $E_{\infty}$ ring spectrum which admits a "six functor formalism" over a suitable class of spaces (by which I mean a context in which what I'm about to say can be made correct).
Let $X$ b …
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