Questions tagged [infinity-topos-theory]

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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 $...
user145752's user avatar
6 votes
0 answers
110 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 ...
Mike Shulman's user avatar
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6 votes
0 answers
301 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 ...
Mike Shulman's user avatar
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5 votes
1 answer
179 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 \...
Mike Shulman's user avatar
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9 votes
0 answers
376 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 ...
Markus Zetto's user avatar
2 votes
1 answer
345 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 (...
Arshak Aivazian's user avatar
2 votes
0 answers
136 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 ...
Tim Campion's user avatar
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8 votes
0 answers
134 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 ...
Tim Campion's user avatar
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4 votes
0 answers
92 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 ...
Tim Campion's user avatar
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3 votes
0 answers
87 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, ...
Andrew Dudzik's user avatar
5 votes
0 answers
233 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 \...
Tim Campion's user avatar
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6 votes
0 answers
123 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 ...
Tim Campion's user avatar
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14 votes
2 answers
557 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 $\mathcal A$ ...
Tim Campion's user avatar
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1 vote
0 answers
175 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 $\infty$-...
CuriousKid7's user avatar
1 vote
0 answers
181 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 ...
Alexander Praehauser's user avatar
6 votes
1 answer
158 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 ...
Tim Campion's user avatar
  • 55.4k
7 votes
1 answer
513 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 ...
Doron Grossman-Naples's user avatar
2 votes
1 answer
319 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 ...
user40276's user avatar
  • 2,139
2 votes
1 answer
171 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 ...
Markus Zetto's user avatar
11 votes
1 answer
512 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 ...
Zhen Lin's user avatar
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12 votes
2 answers
538 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 ...
Zhen Lin's user avatar
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5 votes
1 answer
359 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 ...
André Henriques's user avatar
2 votes
0 answers
204 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 $\Phi [0<1] = \Phi$ (...
Mathemologist's user avatar
6 votes
2 answers
538 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, ...
user90041's user avatar
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7 votes
1 answer
356 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 ...
Tim Campion's user avatar
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7 votes
0 answers
305 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, ...
Tim Campion's user avatar
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9 votes
0 answers
517 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 $\...
Markus Zetto's user avatar
2 votes
1 answer
342 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 ...
Emilio Minichiello's user avatar
1 vote
1 answer
137 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{\...
Emilio Minichiello's user avatar
5 votes
0 answers
300 views

Is the category of Turing categories an ∞-topos?

I'm an independent researcher working on programming languages. I would love to hear that this is all already known by simpler means, or that I'm wrong. I first opened this can of worms in the Before ...
Corbin's user avatar
  • 405
2 votes
1 answer
157 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 ...
Markus Zetto's user avatar
6 votes
1 answer
154 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 ...
Emilio Minichiello's user avatar
2 votes
1 answer
411 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 $n$-...
Markus Zetto's user avatar
15 votes
1 answer
1k 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 $ExDisc_\kappa \to \mathsf{RTop}$ ...
Maxime Ramzi's user avatar
  • 11.2k
13 votes
0 answers
259 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 ...
Tim Campion's user avatar
  • 55.4k
4 votes
1 answer
368 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,...
curious math guy's user avatar
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 ...
David Corfield's user avatar
4 votes
1 answer
325 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 (=...
Tim Campion's user avatar
  • 55.4k
15 votes
1 answer
978 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 ...
Tim Campion's user avatar
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87 votes
10 answers
12k 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. ...
Peter Scholze's user avatar
5 votes
1 answer
162 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 $\infty$-...
Christoph Weis's user avatar
9 votes
1 answer
240 views

Object classifiers in 1-toposes

In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to ...
Giulio Lo Monaco's user avatar
6 votes
0 answers
271 views

Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?

A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model ...
Adittya Chaudhuri's user avatar
5 votes
0 answers
181 views

Surprising examples of functors which preserve cofiltered limits but not all limits?

Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits? I'm not quite sure what "surprising" means, but I think ...
Tim Campion's user avatar
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4 votes
0 answers
177 views

A couple of points in a proof about of $\infty$-toposes

I wanted to have a better understanding of the geometric interpretation of $\infty$-toposes, and in particular learn something about étale morphisms, but I got stuck trying to unravel two points in ...
Giulio Lo Monaco's user avatar
2 votes
1 answer
262 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....
David C's user avatar
  • 9,630
14 votes
0 answers
158 views

Characterization of geometric morphisms without referring explicitly to the left adjoint?

Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
Tim Campion's user avatar
  • 55.4k
8 votes
1 answer
328 views

When is an $\infty$-categorical localization left exact?

Let $L: \mathcal C^\to_\leftarrow L\mathcal C : i$ be an adjunction with $i$ fully faithful. In ordinary category theory, $L$ is left exact iff the class of $L$-local morphisms is stable under base ...
Tim Campion's user avatar
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50 votes
2 answers
3k 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 ...
Charles Rezk's user avatar
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8 votes
1 answer
484 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 ...
Tashi Walde's user avatar