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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.
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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3
votes
1
answer
149
views
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:
$$fiber(Map(X,Y) \to …
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6
votes
1
answer
2k
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(Geometric) Proof for the projective bundle formula in K-theory
I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove:
Projective bundle formula: Let $\pi: E \to X$ be a ve …
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12
votes
1
answer
853
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The (fiber of the) cofiber of the fiber of a map of spaces
Consider a fiber sequence of spaces
$$F \overset{i}{\to} E \to B$$
The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point se …
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5
votes
0
answers
219
views
Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in...
All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
The element …
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4
votes
2
answers
573
views
Principal bundles that can't be detected by spheres
The question I'm trying to answer is the following:
Let $P \to X$ be a principal $G$-bundle (over a connected CW complex)
satisfying that all pullbacks to spheres (of arbitrary dimension) are
…
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9
votes
2
answers
582
views
Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids
Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations.
Generally $Vect(BG) \ne …
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5
votes
1
answer
548
views
Defining hom spaces in the derived category as limits of hom spaces in the homotopy category
Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
A. The following isomorphisms hold:
$$\li …
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9
votes
1
answer
326
views
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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6
votes
1
answer
341
views
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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8
votes
2
answers
533
views
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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5
votes
0
answers
335
views
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 …
12
votes
0
answers
403
views
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
votes
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
votes
0
answers
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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