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Results tagged with homotopy-theory
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user 2191
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.
3
votes
1
answer
195
views
Can one define relative Hurewicz maps using the Dold-Thom theorem
Let $A\to X$ be a (Hurewicz) cofibration of path-connected topological spaces. Then we have a long homotopy sequence $$ \dots\to \pi_i(A)\to \pi_i(X)\to \pi_i(X,A)\to \dots; $$ here we fix a base poin …
- 16.9k
1
vote
When is a thick subcategory the preimage of a weak Serre class under a homological functor?
"Almost certainly" (as far as I remember, one needs either a set of generators for $\mathcal P$ or the Vopenka principle; see Theorem 7.2.1 of Krause's https://arxiv.org/abs/0806.1324) if $\mathcal{P …
- 16.9k
8
votes
Accepted
When does a triangulated category have a heart?
A silly remark is that "trivial" $t$-structures always exist. You should probably say that you want a bounded or a non-degenerate $t$-structure. As far as I remember, non-zero negative $K$-groups of $ …
- 16.9k
5
votes
1
answer
318
views
Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?
Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the homo …
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15
votes
Accepted
Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutat …
- 16.9k
2
votes
1
answer
1k
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Homotopy groups of filtered homotopy limits
Let $X$ be the homotopy limit of a filtered system of simplicial sets $X_i$. When are the morphisms $\pi_j(X)\to \varprojlim \pi_j(X_i)$ surjective for all $j\ge 0$? This seems to be no problem when t …
- 16.9k
7
votes
2
answers
400
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Properness of the category of modules over a spectrum (that represents algebraic cobordism o...
The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak equivalen …
- 16.9k
2
votes
0
answers
184
views
Quillen functors and stable model categories
Are there any books or papers where I can find some general statements and methods for working with Quillen functors that are not equivalences (and not localizations)? In particular, I would like to k …
- 16.9k
4
votes
1
answer
164
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On closed model categories: standard arguments and fibrantly cogenerated categories
Some not very clever questions on closed model categories.
For a (left or right) Quillen functor $F:C\to D$ what arguments does one usually use for proving that $Ho F$ is fully faithful when restric …
- 16.9k
4
votes
1
answer
199
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Yoneda embeddings of stable model categories; composition with Bousfield localizations
For a stable model category $C$ and a set $M$ of object of it I would like to construct a natural functor from $C$ to some stable 'category of functors' on $M$. I suspect that the 'natural' question t …
- 16.9k
4
votes
1
answer
448
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$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?
I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through the …
- 16.9k
5
votes
2
answers
707
views
On triangulated categories of pro-objects
Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) …
- 16.9k
6
votes
1
answer
535
views
Exceptional collections of objects in topological triangulated categories?
People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in …
- 16.9k
9
votes
0
answers
332
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Is it possible to define a perverse $t$-structure for a certain triangulated category of she...
The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (p …
- 16.9k
6
votes
4
answers
644
views
(Co)homological characterization of homotopy pullbacks
For a commutative square of spaces (of manifolds, or of simplicial sets):
$$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & D\end{array}\right)$$ I am tr …
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