All Questions
Tagged with infinity-topos-theory at.algebraic-topology
20
questions
4
votes
1
answer
126
views
Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{...
- 10.3k
7
votes
0
answers
174
views
How does nullification of $K(\mathbb{Z}, 2)$ compare to 1-truncation?
Let $B$ be a type (or space). A type (or space) is $B$-null if the canonical map $X \to X^B$ is an equivalence. The $B$-nullification of an arbitrary type $X$ is a $B$-null type $\bigcirc_B X$ ...
- 3,876
11
votes
1
answer
571
views
What is the connection between Lurie's definition of shape and Čech homotopy?
It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [Higher topos theory] defines this one:
Definition 1.
The ...
- 15k
2
votes
1
answer
279
views
Abelian versions of straightening and unstraightening functors
Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1....
- 9,832
9
votes
1
answer
581
views
Higher-dimensional version of the "Magic Cube Lemma" for homotopy pushouts/pullbacks
The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces:
Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...
- 725
6
votes
0
answers
170
views
Is there an $\infty$-topos of monochromatic spaces?
Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
- 62.6k
14
votes
2
answers
730
views
Examples of topoi that are not ordinary spaces
In [SGA6] we find:
Mais nous lui conseillons néanmoins, de préférence, de s'assimiler le langage des topos, qui fournit un principe d'unification extrêmement commode. (DeepL translate: However, we ...
user144439
14
votes
1
answer
477
views
Comonadicity of spaces over spectra?
As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^...
- 40.8k
6
votes
1
answer
389
views
Understanding model independently the equivalence of two ways of obtaining homotopy types from categories
It is well known that any homotopy type can be obtained as the classifying space of a ($1$-)category. The classifying space of a category $\mathcal{C}$ can be interpreted in at least two ways:
We ...
- 651
5
votes
1
answer
658
views
Associative Ring Spectra and Derived Completion
So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:
Is ...
- 10.3k
7
votes
0
answers
787
views
Commutation of simplicial homotopy colimits and homotopy products in spaces
Edit: The claim below is wrong, as explained in the comments,
because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...
- 37.2k
26
votes
1
answer
1k
views
Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?
It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the $\...
- 9,306
15
votes
2
answers
2k
views
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 ...
- 15k
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 ...
- 15k
9
votes
1
answer
547
views
Homotopy left-exactness of a left derived functor
Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: \mathrm{Ho}(\...
- 3,386