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Results tagged with homotopy-theory
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user 126667
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.
14
votes
Accepted
Do h-coequalizers and coproducts give all h-colimits?
There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over Δop). If $F : I \to M$ is a diagram in a model category, one has
$$\operatorname{hocolim}_I …
- 24.2k
13
votes
"Models" in homotopy theory
Maybe a "lower" analogue would be helpful.
An ordered pair is an object that contains two pieces of data, the first component and the second component.
Suppose we want to make this precise in the lang …
- 25.2k
3
votes
Is an ∞-topos of local homotopy dimension $\leq n$ of homotopy dimension $\leq n$?
As a counterexample to the statement at the end of the question, the $\infty$-topos of parameterized spectra is even of homotopy dimension $\le -1$, but not hypercomplete--as Mathieu Anel pointed out …
- 25.2k
2
votes
Accepted
pair of injective morphisms of simplicial groups
Pick pointed topological spaces $A$ and $B$ which admit pointed injective continuous maps $A \to B$ and $B \to A$ for which $A$ is contractible but $B$ has nonvanishing reduced homology. For example, …
- 25.2k
26
votes
Accepted
Counter-example to the existence of left Bousfield localization of combinatorial model category
A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the stru …
- 25.2k
5
votes
Accepted
What structure of a monoidal simplicial model category is preserved by taking the opposite c...
The general statement is that if $V$ is a monoidal model category (I will assume the unit object of $V$ is cofibrant, so that there are no funny extra axioms related to the unit) and $M$ is a $V$-mode …
- 25.2k
65
votes
Accepted
Analogue to covering space for higher homotopy groups?
There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ fo …
- 19.4k
43
votes
Are there two non-homotopy equivalent spaces with equal homotopy groups?
All of these examples involve a bit of cleverness, so I thought I'd point out a more straightforward way to construct counterexamples. If $X$ is any space, we can build a space $X' = K(\pi_0 X, 0) \t …
- 30.3k
9
votes
Accepted
Homotopy Pushouts via Model Structure in Top
Question 1: The model category $\mathcal{C}$ should be left proper, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper. …
- 30.3k
7
votes
Accepted
Equivalences in Model Categories
Yes. The isomorphism in $\mathrm{Ho}(\mathcal{M})$ is represented by a morphism in $\mathcal{M}$ from a cofibrant replacement for $A$ to a fibrant replacement for $B$. The "converse to the Whitehead …
- 25.2k
9
votes
Accepted
$(\infty,1)$-categories and model categories
Mostly I refer you to my answer here and also this question.
To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a m …
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- 1
18
votes
Accepted
Are non-empty finite sets a Grothendieck test category?
That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left …
- 25.2k
12
votes
Definition of homotopy limits
You can fix it by making η a homotopy coherent diagram (a map to each object of D, a homotopy for each arrow of D, a homotopy-between-homotopies for each commuting triangle in D, ...) and also replaci …
- 25.2k
3
votes
What are the fibrant objects in the injective model structure?
I'm not 100% sure, but I think the answer is that you should choose a cellular model for PSh(C) (the category of presheaves of sets on C), which is a set S of monomorphisms in PSh(C) such that every m …
- 25.2k
13
votes
Why do finite homotopy groups imply finite homology groups?
It sounds like maybe you can prove your first statement for simply connected spaces. In that case, you can use the homotopy orbit spectral sequence (the Serre spectral sequence associated to the fibr …
- 25.2k