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Results tagged with homotopy-theory
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user 11640
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.
19
votes
3
answers
1k
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What are finite homotopy types?
Starting from an ordinary 1-categorical point of view, there are various obvious candidate definitions for ‘finite homotopy type’:
The homotopy type of a simplicial set that has only finitely many n …
- 15.9k
16
votes
1
answer
916
views
The state of the art in the rectification of homotopy-coherent structures
My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of:
Cordier and Porter proved a …
- 15.9k
15
votes
2
answers
2k
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Modern versions of Verdier's hypercovering theorem?
Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a simplicia …
- 15.9k
14
votes
Accepted
Quasicategories for non-simplicial model categories
It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double …
- 15.9k
11
votes
What is a good basic reference on model categories?
Hirschhorn's book, Model categories and their localizations, is a very thorough reference with many basic results explicitly stated and proved. The result you want is implied by axiom SM7 for simplici …
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10
votes
Accepted
How should I be thinking about object classifiers / universal fibrations / universes?
Yes, the universe is the classifying space for small homotopy types. For various reasons, the universe is not itself a small homotopy type; so it fails to be an object classifier for the trivial reaso …
- 15.9k
10
votes
Accepted
Fiber vs homotopy fiber in model categories: simple question
I work in a general pointed model category. The homotopy fibre of a morphism $f : Y \to X$ can be defined as follows: first, choose a fibrant replacement $w_X : X \to \hat{X}$ and then factor $w_X \ci …
- 15.9k
8
votes
Accepted
About the dual of the cube lemma in homotopy theory
Yes, $D_0 \times_{D_1} D_2 \to (E_2 \times_{E_1} E_0) \times_{E_0} D_0$ is a fibration.
First, observe that
$$(E_2 \times_{E_1} E_0) \times_{E_0} D_0 \cong E_2 \times_{E_1} D_0 \cong (E_2 \times_{E_1} …
- 15.9k
8
votes
Accepted
Reedy fibrancy and composition in Segal spaces
Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism
$ …
- 15.9k
7
votes
Modern versions of Verdier's hypercovering theorem?
Here are some remarks on Charles Rezk's answer:
Everything is happening in the category of locally fibrant presheaves, which has a homotopy calculus of right fractions in the sense of Dwyer and Kan …
- 15.9k
6
votes
Contractibility of the category of cosimplicial resolutions
Since you have functorial factorisations you should exploit that to the hilt.
If $\mathcal{M}$ is a model category with functorial factorisations then the category $\mathbf{c}\mathcal{M}$ of cosimplic …
- 15.9k
6
votes
Accepted
Is the composite of absolute derived functors a derived functor?
Here is a somewhat degenerate example that illustrates what can go wrong.
Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the …
- 15.9k
6
votes
0
answers
283
views
What are the Čech-local equivalences of (simplicial pre)sheaves?
Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield …
- 15.9k
5
votes
Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp...
The nerve functor does not preserve homotopy colimits. Indeed, take any simplicial set $X$ with non-trivial $\pi_n$ ($n > 1$) and consider $X$ as a simplicial diagram of sets. In $\mathbf{sSet}$, its …
- 15.9k
4
votes
Accepted
Equivalent definition of a Kan fibration
The class of morphisms having the right lifting property with respect to $\Lambda^1_k \times \Delta^n \hookrightarrow \Delta^1 \times \Delta^n$ (for all $k \in \{ 0, 1 \}$ and all $n \ge 0$) is strict …
- 15.9k