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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.
3
votes
Accepted
Homotopy limits of homotopically constant diagrams over contractible categories
It is true. You can reduce to the case of simplicial sets by using the fact that $\mathbf{R} \mathrm{Hom} (T, -)$ preserves homotopy limits and (allowing $T$ to vary) is jointly conservative. Finding …
- 15.9k
1
vote
Categories of spans from categories of fibrant objects
Yes, one can use functorial factorisation (of weak equivalences with a fixed codomain) to show that two categories of spans are homotopy equivalent is correct.
Actually, the only subtle point I am a …
- 15.9k
2
votes
Accepted
Cartesian products between cofibrant simplicial presheaves
If $\mathcal{C}$ has finite products, then the class of projective cofibrations is also closed under finite products. Indeed, since the cartesian product in the category of simplicial presheaves prese …
- 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
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
2
votes
Does a topological hypercover always have free degeneracies?
Here is a proof for the case where $X$ is a Hausdorff space. Note that each $U_n$ is also Hausdorff in this case.
A standard argument shows that the face operators of $U_\bullet$ are (surjective) loc …
- 15.9k
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
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
$$(E_2 \times_{E_1} E_0) \times_{E_0} D_0 \cong E_2 \times_{E_1} D_0 \cong (E_2 \times_{E_1} …
- 15.9k
4
votes
Accepted
smash product of pointed spaces preserve weak equivalences
Yes, $\wedge$ is invariant under weak homotopy equivalence in $\mathbf{sSet}_*$. It suffices to show that it sends anodyne extensions (= trivial cofibrations) to weak homtoopy equivalences: indeed, ev …
- 15.9k
6
votes
Accepted
Is the composite of absolute derived functors a derived functor?
Here is a somewhat degenerate example that illustrates what can go wrong.
Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the …
- 15.9k
8
votes
Accepted
Reedy fibrancy and composition in Segal spaces
Given a simplicial object $X$ in a locally small category $\mathcal{M}$ and a simplicial set $S$, define the weighted limit $\{ S, X \}$ to be an object in $\mathcal{M}$ equipped with an isomorphism
$ …
- 15.9k
2
votes
DK equivalences are Reedy equivalences for complete Segal spaces
Let $U$ and $V$ be Segal spaces and let $f : U \to V$ be a Dwyer–Kan equivalence.
That means two things:
The following diagram is a homotopy pullback square:
$$\require{AMScd}
\begin{CD}
U_1 @>>> U_0 …
- 15.9k
19
votes
3
answers
1k
views
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
4
votes
2
answers
453
views
Aspheric functors and Grothendieck fibrations
Following Grothendieck, let us say that a category is aspheric if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is aspheric if for every object $b$ in $\mathcal{B}$, …
- 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