# Questions tagged [infinity-topos-theory]

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

112
questions

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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 ...

7
votes

1
answer

385
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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 ...

2
votes

1
answer

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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 ...

2
votes

1
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147
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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 ...

10
votes

1
answer

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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 ...

11
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2
answers

409
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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 ...

5
votes

1
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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 ...

2
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0
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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 $\Phi [0<1] = \Phi$ (...

5
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2
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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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1
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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 ...

7
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0
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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, ...

9
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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 $\...

1
vote

1
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248
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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 ...

1
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1
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118
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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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252
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### 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 ...

2
votes

1
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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 ...

5
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1
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141
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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 ...

2
votes

1
answer

363
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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 $n$-...

13
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1
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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 ...

4
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1
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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 ...

4
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1
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287
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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 (=...

14
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1
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### 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 ...

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votes

10
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### 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. ...

4
votes

1
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### 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 $\infty$-...

9
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1
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### 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 ...

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### 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 ...

5
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156
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### 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 ...

4
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0
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### 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 ...

2
votes

1
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242
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### 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....

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### 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 ...

7
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1
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292
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### 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 ...

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### 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 ...

8
votes

1
answer

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### 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 ...

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### 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 ...

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1
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### Are constant $\infty$-sheaves constant on connected components?

Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose ...

1
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0
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182
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### A categorial PCF theory?

I'm not an expert in PCF theory, so please forgive me if this question makes no sense.
I'm looking for a categorial version of PCF theory.
Specifically, if we replace $Set$ with another category, ...

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2
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### 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 ...

5
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1
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### Are semisimplicial hypercoverings in a hypercomplete $\infty$-topos effective?

A hypercovering in an $\infty$-topos $E$ is a simplicial object $U \in {\rm Fun}(\Delta^{\rm op},E)$ such that each map $U_n \to ({\rm cosk}_{n-1} U)_n$ is an effective epimorphism. Theorem 6.5.3.12 ...

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### 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 ...

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2
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### Commutative rings : Topoi = Fields :?

The following is probably a bad question, but hopefully, it might have a very good answer.
In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...

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1
answer

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### Stability of accessible $\infty$-categories under some operations

I'm reading through Higher Topos Theory, and I can't make sense of a few proofs in the sections about accessible $\infty$-categories.
In Proposition 5.4.4.3, Lemma 5.4.4.2 is used, but I don't see ...

18
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1
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### Equivalences of categories of sheaves vs categories of $\infty$-Stack

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.
$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$
And we want to ...

4
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1
answer

234
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### A finite Whitehead Theorem for $\infty$-topos

Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces ...

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591
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### What is a spectrum object in $\infty$-topoi?

For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...

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1
answer

701
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### What are the monomorphisms of ($\infty$-)toposes?

There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...

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1
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### Are there continua in $\infty$-topoi?

If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....

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### What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?

Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...

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### Internal van Kampen colimits

Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...