All Questions
Tagged with topos-theory homotopy-theory
25 questions
6
votes
0
answers
188
views
The shape of an ($\infty$-)topos as a monad
Let $\mathcal E$ be an $\infty$-topos, and let $t : \mathcal E \to Spaces$ be the unique geometric morphism to $Spaces$. The composite $t_\ast t^\ast : Spaces \to Spaces$ is a left-exact accessible ...
- 63.9k
1
vote
0
answers
214
views
Kan extensions in Grothendieck school
Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
- 894
9
votes
1
answer
584
views
What is known about the homotopy type of the classifier of subobjects of simplicial sets?
For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that
For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$
For $...
- 6,962
3
votes
1
answer
551
views
One-point compactification of a condensed set
Is there a notion of a 'one-point compactification of a condensed set'?
$\textbf{Motivation:}$ For a locally compact space $X$, there is a notion of maps that vanish at infinity. A continuous function ...
- 1,485
36
votes
3
answers
7k
views
Higher Topos Theory- what's the moral?
I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
5
votes
1
answer
284
views
Axioms of derivators
I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute ...
- 894
14
votes
2
answers
693
views
Who introduced the notion of 2-categories?
Wikipedia seems to have an answer
"The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak ...
- 894
9
votes
1
answer
257
views
Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?
To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X\...
- 18.4k
2
votes
0
answers
504
views
Relative homology in Fargues-Scholze paper
if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...
- 1,093
7
votes
2
answers
321
views
Indexing categories of derivators
It is not clear to me the role of the domain and target in the definition of prederivators.
For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself.
...
- 894
11
votes
1
answer
680
views
A non-conventional definition of topoi
In "Toward a Galoisian interpretation of homotopy theory" (2000), B. Toën wrote:
Pour expliquer notre point de vue sur la notion de champs rappelons une construction (non conventionnelle) ...
- 894
8
votes
1
answer
454
views
Is every conservative, left exact left adjoint comonadic, $\infty$-categorically?
Consider a conservative left adjoint $G : C \to D$ between complete 1-categories. By Beck's theorem, the following are equivalent:
$G$ is comonadic.
$G$ preserves $G$-split equalizers.
(2) is ...
- 63.9k
10
votes
1
answer
2k
views
Computations in condensed mathematics, page 32-34
I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:
At the last line of pg 32 - it seems to imply that ...
- 661
52
votes
2
answers
4k
views
What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?
My question is as in the title:
Does anyone have an example (supposing one exists) of an
$\infty$-topos which is known not to be equivalent to sheaves on a
Grothendieck site?
An $\infty$-topos is as ...
- 27.2k
3
votes
1
answer
356
views
Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers
Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
- 365
5
votes
0
answers
442
views
Roadmap to understand gerbe in the sense of Lurie’s Higher Topos Theory
Definition $7.2.2.20$ : Let $\mathfrak{X} $ be an $\infty$-topos. An $n$-gerbe on $\mathfrak{X}$ is an object in $\mathfrak{X}$ which is $n$-connective and $n$-truncated.
Above is the definition of ...
- 6,023
5
votes
1
answer
357
views
Geometric realization of the mapping stack
Some background and notation
Let $Sh_{\infty}(Cartsp)$ be the infinity category of smooth simplicial sheaves on the site of cartesian spaces (convex open subsets of $\mathbb{R}^n$ and smooth maps ...
- 928
11
votes
1
answer
819
views
What is an Elementary "Homotopy, Model" Topos?
Context:
Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C.
Thm (...
- 1,720
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 ...
- 15.9k
10
votes
3
answers
962
views
classifying $\infty$-toposes for topological/localic groups?
Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...
- 42.4k
10
votes
0
answers
913
views
Interaction petit topos - gros topos
I know that there are already several discussions on this topic (and they already helped me a lot, so far), but I couldn't find one completely solving my problem.
Question 1. Fix a scheme $X$. I know ...
- 513
3
votes
0
answers
260
views
A topos-theoretic thickening of the nerve functor
Let $\Delta$ be the standard cosimplicial space sending $[n]\mapsto \Delta^n\subset \mathbb R^{n+1}$.
Then the correspondence $\delta\colon [n]\mapsto {\rm Sh}(\Delta^n)$ defines a cosimplicial ...
- 13.6k
39
votes
4
answers
6k
views
What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?
In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
- 8,512
18
votes
2
answers
1k
views
Is there a topos theoretic interpretation/proof of Quillen's Theorem A?
I think the title says it all. Quillen's Theorem A says that a functor $F\colon C\to D$ induces a homotopy equivalence of classifying spaces if each fiber category $F/d$ with $d$ an object of $D$ is ...
- 38.6k
55
votes
3
answers
5k
views
What are the higher homotopy groups of Spec Z ?
The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
- 8,004