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Results tagged with homotopy-theory
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user 22810
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.
21
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2
answers
2k
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How to stop worrying about enriched categories?
Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they a …
14
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2
answers
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:
$$H^{\bullet}( …
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12
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0
answers
403
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The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from tha...
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the h …
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12
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1
answer
853
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The (fiber of the) cofiber of the fiber of a map of spaces
Consider a fiber sequence of spaces
$$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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9
votes
2
answers
582
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Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids
Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations.
Generally $Vect(BG) \ne …
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9
votes
1
answer
326
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Closed formulas for topological K-theory?
Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line bundl …
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8
votes
2
answers
533
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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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8
votes
1
answer
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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8
votes
1
answer
693
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Simple characterization of Postnikov & Whitehead towers?
I'm asking this question in the most model-ambiguous way I can since this is the kind of answer i'm looking for.
There are various explicit constructions of the Whitehead and Postnikov towers. I'm try …
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7
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0
answers
404
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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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6
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0
answers
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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6
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308
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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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6
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2
answers
626
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Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories
Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions i …
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6
votes
0
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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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6
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4
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What does "higher monodromy" tell us about a principal bundle
Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to con …
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