Questions tagged [infinity-topos-theory]
The infinity-topos-theory tag has no usage guidance.
136
questions
3
votes
1
answer
134
views
Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?
Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits.
My ...
6
votes
1
answer
306
views
$\infty$-topos as an internal $\infty$-category in itself
I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...
8
votes
0
answers
219
views
What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?
The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ .
The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{...
5
votes
1
answer
252
views
Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...
2
votes
0
answers
58
views
Coequalizers and pullbacks in $\infty$-topoi
In an $\infty$-topos, suppose we have two cartesian diagrams of the form
$$
\require{AMScd}
\begin{CD}
\overline{A} @>>> \overline{B} \\
@VVV @VVV \\
A @>>> B .
\end{CD}
$$
Let
$$
\...
8
votes
0
answers
387
views
Sheaf of compact Hausdorff spaces but not a condensed anima
Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
10
votes
2
answers
565
views
Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)
In Proposition 7.2.1.14 of Higher Topos Theory, Lurie asserts the following:
Let $\mathcal{X}$ be an $\infty$-topos and let $\tau_{\leq0}:\mathcal{X}\to\tau_{\leq0}\mathcal{X}$ denote a left adjoint ...
4
votes
2
answers
449
views
Categorical equivalences vs. categories of simplices
Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures)
$$
j_!:\mathsf{sSet}_{/K}\...
5
votes
1
answer
213
views
Connectedness of truncated version of cosimplicial indexing category
Let $F:\mathbf{\Delta}\to\mathcal{S}_{\leqslant n-1}$ be a cosimplicial object in the $\infty$-category of $(n-1)$-truncated spaces. Is it always a right Kan extension of its restriction along $\...
7
votes
0
answers
166
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 $\bigcirc_B X$ ...
2
votes
1
answer
87
views
Comparion theorem between symmetric monoidal $\infty$-functor
Let $T,T'$ be symmetric monoidal $\infty$-categories. And let $F_1,F_2:T\to T'$ be symmetric monoidal functors and let $t:F_1\Longrightarrow F_2$ be a symmetric monoidal natural transformation from $...
7
votes
0
answers
143
views
Presenting geometric morphisms by geometric morphisms
It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a ...
6
votes
0
answers
337
views
$(\infty,1)$-topoi generated by $(n,1)$-categories
A (1,1)-topos (i.e. an ordinary Grothendieck topos) is called localic if the following two equivalent conditions hold:
It is the category of sheaves on a (0,1)-site with finite limits$^*$ (i.e. a ...
5
votes
1
answer
195
views
Fibrations of sites for $\infty$-topoi
For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \...
9
votes
0
answers
438
views
Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
3
votes
1
answer
411
views
Does the concept of a $\infty$-category have a natural definition in the $\infty$-world?
I start with a thesis: the natural notion of equality is additional data (paths/morphisms), not a binary relation (the fact that they exist). So, in particular, with such a constructivization (...
2
votes
0
answers
158
views
Is every sheaf $\infty$-topos equivalent to sheaves on itself with respect to the canonical topology?
Let $(\mathcal C, J)$ be a small subcanonical $\infty$-site, and let $Sh_J(\mathcal C)$ be the $\infty$-topos of sheaves thereon. Then $Sh_J(\mathcal C)$ is itself an $\infty$-site with respect to the ...
8
votes
0
answers
142
views
The tangent bundle and dual tangent bundle in topos theory
Let $\mathcal B = B \mathbb T$ be an $\infty$-topos, thought of as the classifying $\infty$-topos of some "$\infty$-geometric theory" $\mathbb T$. The notion of "$\infty$-geometric ...
4
votes
0
answers
104
views
The hyperdoctrine of topoi and a near KZ-comonad
Allow me to waffle about a bit of topos theory, leading up to a few questions about KZ comonads and about the comprehension schema in hyperdoctrines.
Let $Pr^L$ denote the $\infty$-category of ...
3
votes
0
answers
94
views
When do geometric morphisms lead to periodic adjoints?
This may be a naïve question but I've been unable to locate a reference that addresses it. Any thoughts are appreciated!
Let $f:\mathcal{E}\to\mathcal{S}$ be a cohesive morphism of toposes. That is, ...
5
votes
0
answers
243
views
Do topoi have injective hulls?
Recall that a topos $\mathcal I$ is injective (with respect to embeddings) if and only if $\mathcal I$ is a retract of $Psh(C)$ for some finitely-complete $C$. Say that an embedding $f : \mathcal X \...
6
votes
0
answers
152
views
Does $Pr^L_\kappa \in Pr^L$ behave like an "object classifier" or "universe"?
Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories and left adjoint functors. Let $Pr^L_\kappa \subset Pr^L$ denote the locally-full subcategory of $\kappa$-compactly-generated ...
14
votes
2
answers
656
views
When is a stable $\infty$-category the stabilization of an $\infty$-topos?
Let $\mathcal X$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal X)$ of $\mathcal X$ is the universal presentable stable category on $\mathcal X$.
Conversely, if $\mathcal A$ ...
1
vote
0
answers
184
views
Is there a (Grothendieck) $\infty$-topos for which Whitehead's theorem only holds for maps between truncated objects?
We know that non-hypercomplete $\infty$-toposes exist. Is there such a topos $\mathcal{E}$ with the following property?
For any $X, Y \in \mathcal{E}$, if all weak homotopy
equivalences (or $\infty$-...
1
vote
0
answers
230
views
Understanding the double negation modality under the "propositions as types" paradigm
$\DeclareMathOperator\Hom{Hom}$I'm trying to understand the double negation modality under the "propositions as types" paradigm, but I'm running into an apparent contradiction: let $T$ be a ...
8
votes
1
answer
268
views
Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?
$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...
7
votes
1
answer
563
views
If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?
I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...
2
votes
1
answer
356
views
Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$
$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
2
votes
1
answer
183
views
Subcategory of coherent objects in an $\infty$-topos forming a local $\infty$-pretopos
My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the ...
11
votes
1
answer
562
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 ...
15
votes
2
answers
633
views
How to formulate the univalence axiom without universes?
The standard formulation of the univalence axiom for a universe type $U$ is that, for all $X : U$ and $Y : U$, the canonical map $(X =_U Y) \to (X \simeq Y)$ is an equivalence.
As we (usually) cannot ...
5
votes
1
answer
396
views
Geometric realisation of smooth $\infty$-stacks
Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial ...
2
votes
0
answers
207
views
A map that names itself
Call the walking arrow $\Delta_{1}$, containing exactly one nontrivial 1-cell $[0<1] : 0 \to 1$. I am interested in a map $\Phi : \Delta_{1} \to \mathrm{Type}$, such that $\Phi [0<1] = \Phi$ (...
6
votes
2
answers
799
views
Learning roadmap to 'Differential cohomology in a cohesive $\infty$ topos'
I am very curious to study arXiv:1310.7930 (henceforth:DCCT) but am not sure if I have the pre-requisites. I am familiar with basic algebraic topology (singular cohomology, classifying spaces, ...
7
votes
1
answer
413
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 ...
7
votes
0
answers
339
views
Does every exponentiable ($\infty$-)topos have enough points?
The notion of a coherent topos is a somewhat "refined" finiteness condition to put on a topos. One reason it is considered a "fruitful" notion is the Deligne completeness theorem, ...
9
votes
0
answers
555
views
Coherent objects in a hypercomplete $\infty$-topos
In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...
2
votes
1
answer
443
views
Equivalence relations, Segal groupoids and groupoid objects in an infinity category
There are three forms of "equivalence relations are effective" as part of Giraud's axioms in $1$-Grothendieck topoi, Model topoi and Infinity topoi. I am trying to understand how they relate ...
1
vote
1
answer
169
views
n-truncation/n-connected factorization in an $\infty$-topoi
I want to prove that given an $\infty$-topos $\mathscr{C}$ and a morphism $f: X \to Y$ in $\mathscr{C}$, for each $k \geq -1$, there exists a factorization $X \xrightarrow{\eta_f} E_k^f \xrightarrow{\...
2
votes
1
answer
165
views
(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces
I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a ...
6
votes
1
answer
166
views
Join as a bifunctor
I have been reading these great notes by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz ...
2
votes
1
answer
440
views
Counterexamples concerning $\infty$-topoi with infinite homotopy dimension
In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:
Homotopy dimension (henceforth h.dim.), which is $\leq n$ if $n$-...
15
votes
1
answer
2k
views
$\infty$-topoi versus condensed anima
Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ ...
13
votes
0
answers
268
views
Which topoi are local with respect to Stone-Cech compactification?
Compact Hausdorff spaces $X$ are characterized among all topological spaces by the fact that for any topological space $S$, the embedding $S \to \beta S$ into its Stone-Cech compactification induces a ...
4
votes
1
answer
416
views
Do stalks see epimorphism of stacks?
Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to https://mathoverflow.net/q/307366, this is an epimorphism if and only if it is almost surjective,...
9
votes
1
answer
1k
views
Cohesion relative to a pyknotic/condensed base
Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial ...
4
votes
1
answer
349
views
Is $Set$ a tiny topos?
Let $Topos$ be the $(2,1)$-category of Grothendieck toposes and geometric morphisms. This is a $V$-sized, locally $V$-sized, locally locally small $(2,1)$-category with all small (2,1)-colimits (=...
15
votes
1
answer
1k
views
Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos?
Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow ...
91
votes
10
answers
14k
views
Reflection principle vs universes
In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
5
votes
1
answer
176
views
Computing cohomology using bounded hypercovers
Let $G$ be a Lie group (paracompact, not necessarily compact), and $A$ an abelian Lie group. I want to write down cocycles in $\mathrm{H^n}(\mathbf{B}G,A)$, the cohomology in the cohesive $\infty$-...