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Results tagged with homotopy-theory
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user 12547
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.
6
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0
answers
208
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Can representable presheaves be made injectively fibrant?
I suspect that the answer to my question is no, but let me give it a shot anyway.
If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet} …
- 85
19
votes
2
answers
2k
views
Does the bordism homology theory satisfy the weak equivalence axiom?
There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is …
- 5,829
7
votes
Accepted
Do finite simplicial sets jointly detect isomorphisms in the homotopy category?
The answer is no. Otherwise, it would follow from Brown's representability theorem (and here I mean very specifically Theorem 2.8 from Brown's 1965 paper Abstract Homotopy Theory) that every "half-exa …
- 7,656
6
votes
Accepted
Homotopy pushout independent of factorization and symmetric in cofibration category
I don't remember how Baues does this exactly, but all facts of this sort follow from the Gluing Lemma (see Lemma 1.4.1 in this paper) and "Brown type factorization". By this I mean the following const …
- 7,656
5
votes
Functors between simplicial sets and cubical sets with connections
I believe that the standard barycentric subdivision functor $\mathrm{Sd} \colon \mathsf{sSet} \to \mathsf{sSet}$ factors through cubical sets with connections. We define a functor $S \colon \Delta \to …
- 7,656
6
votes
Accepted
Simple characterization of Postnikov & Whitehead towers?
Indeed, the properties you stated characterize Postnikov and Whitehead towers. A nice conceptual way of justifying this is by using $k$-connected / $k-$truncated factorization systems.
To fix termino …
- 7,656
2
votes
Accepted
Geometry of the second barycentric subdivision (and Thomason-fibrant replacement)
Yes, that's true. See Remark 4.1 in this paper by Meier and Ozornova. A generalization to $k > 2$ would be interesting and I believe that everything you wrote about it is right. However, I don't know …
- 7,656
16
votes
Accepted
On a weaker version of homotopy equivalence between topological spaces
Saying that there are maps $f \colon X \to Y$ and $g \colon Y \to X$ such that $g f$ is homotopic to $\mathrm{id}_X$ means that $X$ is a homotopy retract of $Y$. (By the way, we say that maps are homo …
- 7,656
8
votes
Accepted
Homotopy pullbacks/relative homotopy groups vs homotopy pushouts/relative homology groups
Relative homotopy groups are the homotopy groups of the homotopy fiber. A homotopy pullback square induces an equivalence of the homotopy fibers of two of its parellel maps by the cancellation propert …
- 7,656
2
votes
Ref request: making a homotopy equivalence fibrewise, in an abstract setting (eg fibration c...
I was quite sure that this was explicitly written down by Rădulescu-Banu or by Hirschhorn, but I cannot find this exact statement.
However, this follows directly by combining Theorems 7.5.10 and 7.6. …
- 7,656
4
votes
Accepted
Saturated classes and cofibrantly generated model structures
It doesn't make a difference as long as we restrict attention to compactly generated saturated classes, i.e. cofibrantly generated ones where generators can be chosen to have $\aleph_0$-small domains. …
- 7,656
2
votes
Cube Lemma on a cofibrantly generated (almost) model category
This is not an answer in a full generality, but it is certainly not the case if the domains of generating acyclic cofibrations are cofibrant. (I have a hard time thinking of an example of a model cate …
- 7,656
10
votes
Accepted
Can a weak fibration category be non saturated?
It is a result of Cisinski that in a fibration category the three conditions you mention (saturation, 2-out-of-6, weak equivalences closed under retracts) are all equivalent. See Theorem 7.2.7 in this …
- 7,656
12
votes
Accepted
When do colimits agree with homotopy colimits?
I don't think we can expect to have one general answer to this question, only a collection of unrelated specialized results. Here are two more:
In the category of simplicial sets all filtered colimi …
- 7,656
10
votes
Accepted
Relative version of Quillen's theorem A
The condition that appears in the assumption of what you call "Relative Theorem A" was introduced by Grothendieck in Pursuing Stacks. It is a part of the definition of a basic localizer, i.e. a class …
- 7,656