# Questions tagged [topos-theory]

A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

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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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### Topos of sheaves on a scheme considered as a functor

The spectrum of a ring $R$ can be defined as $\operatorname{Spec} R := \operatorname{Hom}(R, -)\colon \mathrm{fpRing} \to \mathrm{Set}$ ($\mathrm{fpRing}$ are commutative finitely presentable rings). ...

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### G-topological spaces and locales

Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...

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### Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He ...

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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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### Topos semantics of constructive higher order logic

I would like to find a reference that describes the semantics of constructive higher order logic with function types in toposes. In particular, it seems that if we are to take function types as ...

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### Relationship between coarse objects, separated objects, and sheaves

I would like to better understand the relationship between quasitopoi and topoi. Here are two relationships that I am aware of:
Given a local topos $E \to S$, i.e. such that $S$ is equivalent to the ...

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### 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 $...

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### Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions.
Laurent Lafforgue applying Olivia Caramello thesis described in ...

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### Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?

Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...

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### Condition for an equivalence of functor categories to imply an equivalence of categories

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...

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### How to read the definition of Grothendieck Pretopology in SGA4?

In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$
sont quarrables. (Rappelons qu’un morphisme ...

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### Do presheaf toposes satisfy the full fan theorem?

Presheaf toposes satisfy LPO and (edit: if over categories with binary products) PAx and countable choice internally, so they automatically satisfy the stable fan theorem (every bar which is the ...

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### Anafunctors vs the plus construction

Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations
$$G(M) := \text{...

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### Local isomorphism of condensed sets and étale condensed groupoids

Is there a notion of local isomorphism for condensed sets?
$\textbf{Motivation:}$ I am trying to define what an étale condensed groupoid would be.
A topological groupoid $\mathcal{G}$ is said to be ...

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### Constructive theory of Lie algebras

I'm looking for references on constructive Lie algebra theory, e.g. the sort of theory you could develop in Martin-Löf type theory or internal to some topos with a NNO. Obviously excluded middle is ...

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### A kind of “weak” filtered colimit in the effective topos

I was recently reminded that even filtered colimits in the effective topos generally do not exist. However, there is an important (albeit restrictive) situation that looks a lot like them and that I ...

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### (When) do filtered colimits exist in the effective topos?

(My apologies if this is well-known: I feel that I'm missing something very obvious here.)
Basic question: Do filtered colimits exist in the effective topos?
The reason I feel I'm missing something ...

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### Tensor product of sites

Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small ...

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### 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 ...

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### Does the Zariski spectrum of a ring arise formally from the inclusion of the big Zariski topos into the classifying topos for rings?

Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
...

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### Can we encode a torsor as a binary function on the isomorphism classes of objects?

Let $G$ be a group object in a topos $\mathcal{T}$. Then we have the notion of a $G$-torsor in $\mathcal{T}$, and the set of isomorphism classes of such objects is denoted $H^1(\mathcal{T};G)$. For ...

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### Recommendations to learn about the use of toposes in logic?

I'd like to learn about the use of toposes in logic. The "logic" side I know quite well, but of the "topos" side I am totally ignorant.
Which books/articles (formal and/or casual) ...

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### Do Grothendieck topoi with enough points satisfy the fan theorem internally?

Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it.
This leads one to conjecture a ...

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### Cohesive structure of Cahiers and Dubuc topoi

The inclusion of commutative rings into supercommutative rings has two adjoints, one projecting out the even part and the other quotienting out the ideal generated by odd elements. After passing to ...

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### Do all toposes satisfy the internal Zorn's lemma?

I came up with this question when trying to give a more detailed answer to a question by Tim Campion in a comment to Ingo Blechschmidt's answer to Examples of statements that are valid in every ...

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### Topos with $\Omega = [0,1]$?

For weakly cohesive toposes, there exists a notion of contractability, and toposes with a subobject classifier $\Omega$ that is contractible are of special interest (see here).
It occured to me that ...

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### Are there good criteria for the topological models where BD-N and BD hold?

A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have
$\lim_{n\to \infty} \frac{x_n}{n} = 0$
Clearly all bounded subsets are pseudo-...

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### Property of pushouts in the category of unital $C^{\ast}$-algebras

Let $A$ be a unital $C^{\ast}$-algebra and $\{ f_i: A \rightarrow A_i \}_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod_i A_i$ is ...

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### Noncommutative condensed sets

Ignoring set-theoretic problems, we can see condensed sets as sheaves of compact Hausdorff spaces. Using Gelfand Duality we obtain an equivalence of categories
\begin{align*} \mathrm{CHaus}^{\mathrm{...

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### Phenomena of topos

These days I am wandering on a wild adventure in an incredible but intimidating land. Fortunately, I could find a guide to some animals of this land
Phenomena of gerbes
But someone said to me that ...

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### Subobject classifier for sheaves on large sites with WISC

Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...

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### Does a tight apartness relation on a subobject classifier imply the elementary topos is Boolean?

Given a set $S$, a tight apartness relation on $S$ is a relation $\#$ which is tight, irreflexive, symmetric, and a comparison, or more specifically, a relation $\#$ such that
for all elements $a \in ...

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### A reference on a result by Steve Schanuel

In the Author Commentary section of the TAC reprint of the paper of 1968 Diagonal arguments and cartesian closed categories., Bill Lawvere wrote:
‘Nilpotent infinitesimals fall far short of even one-...

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### Right transferred model structure on the category of algebras in the Grothendieck topos

Let M be a model category that is a Grothendieck topos. Let $T$ be a multisorted finitary algebraic theory. Does there exist the right transferred model structure on the category of $T$-algebras in $M$...

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### Condensed vs pyknotic vs consequential

As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications ...

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### Dissolution of a topos

The dissolution of the locale associated with a frame $F$ is the locale associated with the frame $N(F)$ of nuclei of $F$ (see, e.g., Johnstone, “Stone Spaces” (1982), §2.5). Note that there is a ...

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### Properties of pyknotic sets

In Peter Johnstone's 1979 paper On a topological topos, he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}_\infty$, the one-point ...

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### Intrinsic topology on the Zariski spectrum

In a big topos whose objects are a kind of "space", it sometimes happens that when we define some "set" internally to the topos, the "topology" it automatically acquires ...

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### Beck-Chevalley conditions for the local geometric morphisms $\pi:\mathrm{Zar}/X\to \mathrm{Sh}(X)$

$\newcommand{\Zar}{\mathrm{Zar}}\newcommand{\Sh}{\mathrm{Sh}}$The category of schemes is a full subcategory of the big Zariski topos $\Zar$. For each scheme $X$, there is a local geometric morphism $\...

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### Is there a topos of quotients of sets?

The objects of the desired category are epimorphisms of sets $E \to B$ (in what follows, the notation $E/B$ will be used instead of the arrow). Is it possible to naturally define morphisms such that:
...

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### Is the category of modules over a commutative ring the category of abelian objects in a topos?

The categories of modules over commutative rings are especially notable Abelian categories. Wanting to extend this class a bit, I thought of this question:
Let $R$ be a commutative ring with $1$. Does ...

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### Clarification on the relationship of dream mathematics to ZFC and its potential as a synthetic measure theory

I'm interested in dream mathematics (https://ncatlab.org/nlab/show/dream+mathematics) as a foundation of "synthetic measure theory" in a similar vein as synthetic differential geometry, but ...

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### Localic or topos-theoretic definition of $\operatorname{Spec}$

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes&...

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### The Grothendieck topology of closed immersions on schemes

Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...

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### Are the models of infinitesimal analysis (philosophically) circular?

Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful ...

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### Analogue of Kock-Lawvere axiom for power series rings?

The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism
...

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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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### Decategorifying Grothendieck topoi and categorifying topological spaces

(This is in a sense a follow-up to this question.)
I was under the impression these days that Grothendieck topoi were also¹ analogous to topological spaces in that the former were left exact ...

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### Has an uncomputable variant of the Cantor staircase ever been used in constructive logic?

An open problem in choiceless constructivism is to prove that if a function $f:\mathbb R \to \mathbb R$ is pointwise differentiable everywhere, with $f'=0$, then $f$ is constant. See In choiceless ...