# Questions tagged [infinity-topos-theory]

The infinity-topos-theory tag has no usage guidance.

139
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### Elementary ∞-topos theory [closed]

What is elementary higher topos?

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### Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)

Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...

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### "$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)

In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim:
($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...

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### 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 ...

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### $\infty$-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 ...

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### 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{...

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### 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 ...

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### 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
$$
\...

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### 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{...

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### 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 ...

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### 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
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### 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 $\...

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### 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 $\bigcirc_B X$ ...

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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 $...

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### 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 ...

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### $(\infty,1)$-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 ...

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1
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### 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 \...

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### 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 ...

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votes

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 \...

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### 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 ...

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### 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 $\mathcal A$ ...

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### 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 $\infty$-...

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0
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\Phi [0<1] = \Phi$ (...

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### 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, ...

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### 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 ...

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### 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, ...

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### 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 $\...

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### 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 ...

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### 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{\...

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### (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 ...

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### 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 ...

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### 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 $n$-...

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### $\infty$-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 $ExDisc_\kappa \to \mathsf{RTop}$ ...

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### 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 ...

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### 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,...

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### 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 ...

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### 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 (=...