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Results tagged with homotopy-theory
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user 11540
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
Cube Lemma on a cofibrantly generated (almost) model category
If I understand your situation correctly, the answer is yes, the cube lemma holds, but you cannot use that to get a full model structure. Your situation often arises when trying to transfer a model st …
- 30.3k
3
votes
Accepted
Local injective model structure for simplicial presheaves
Many statements in the question are incorrect, and so I want to put an answer here to prevent future visitors to this thread from being confused. First, it is NOT TRUE that localizing at the local wea …
- 30.3k
2
votes
Accepted
Top - Model structures
A different way of getting a model structure which captures "truncation" data was asked about here, and I answered it. There a $\pi_n$ weak equivalence is an isomorphism for $t\leq n$ but we make no m …
- 30.3k
2
votes
mapping spaces of diagrams
I think the best way to go about this is to go through the Reedy model category structure, but for that to work $A$ has to be a Reedy category. If you are in this situation, then Theorem 15.6.4 of Hir …
- 30.3k
4
votes
Alternate proofs of Quillen's theorem on formal group laws and MU
Jacob Lurie taught a course at Harvard in the spring of 2010 which included a slick proof of Quillen's theorem. I'm pretty sure Steenrod operations are not mentioned at all. The course page is here, a …
- 30.3k
4
votes
Are the fibrations between $\mathbb{A}^{1}$-local objects $\mathbb{A}^{1}$-fibration?
Yes. This is a general property of left Bousfield localization. A map between local objects is a fibration (i.e. in $\mathcal{M}$) if and only if it's a local fibration (i.e. fibration in $L_{\mathcal …
- 30.3k
3
votes
Pushout of spaces
When I see questions like this, I always reach for Dwyer-Spalinski "Homotopy theories and model categories." I just saw another question like this a couple of days ago, and a counterexample was given …
- 30.3k
7
votes
Commutativity up to homotopy implies strict commutativity, for lifting problems
I believe the answer is yes. The kind of lift you're asking about was studied extensively in the paper "On Fibrant objects in model categories" by Valery Isaev. Apply Proposition 3.4, with $I = \{A \t …
- 30.3k
4
votes
"Models" in homotopy theory
It is not related to model categories. It's related to mathematical modeling. The point is that we have many possible choices for what we mean by "space." You might mean "topological space" but someon …
- 30.3k
4
votes
Accepted
Excellent monoidal model categories admit enriched fibrant replacement functors?
I think the answer is yes (to both questions). In Emily Riehl's book "Categorical homotopy theory", chapter 13 is all about the enriched small object argument. Theorem 13.2.1 on page 177 (I hope I'm l …
- 30.3k
3
votes
Accepted
Alternative characterization of homotopy equivalence
EDIT: Now that the OP has edited his question to make clearer what he wants as an answer, I'm removing speculation about what he wanted. The answer is: yes, you can characterize homotopy equivalences …
- 30.3k
3
votes
Homotopy pullbacks and pushouts of spectra
Although the answer is sketched in the comments, I wanted to remark that this statement is proved carefully by Cary Malkiewich as Proposition 6.2.11 in Parameterized Spectra, A Low Tech Approach. He c …
- 30.3k
2
votes
Reference for choosing a path lifting function?
Perhaps what you're referring to is Section 7.2 (page 57 of the pdf) of Peter May's A Concise Course in Algebraic Topology. Here he characterizes what it means to be a Hurewicz fibration in terms of p …
- 30.3k
2
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Accepted
Model structures on diagrams indexed by a Reedy category
Your last paragraph is correct. For any ring $R$, Ch(R) is combinatorial because it's a Grothendieck category. Also, the injective and projective model structures on Ch(R) are Quillen equivalent. If $ …
- 30.3k
5
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In CGWH, is every cofibration an inclusion with closed image?
Yes, see Lemma 2.4.5 and Corollary 2.4.6 in Hovey's book Model Categories. This is in the category of all spaces. It's also true for CGWH as is stated in a proof at the top of page 59 of Hovey's book. …
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