Questions tagged [topos-theory]

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3
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1answer
123 views

Sheaves on sites given by a (regular) cd-structure

Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...
7
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1answer
199 views

When do two topoi have the same cohomology of constant sheaves

Recently, I have some questions for some generalizations from algebraic topology. I learn some homotopy theory in algebraic topology. We know that, if two spaces are homotopy, then they have same ...
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0answers
159 views

How much is known about the consistency strength of toposes and topos-like categories?

It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
14
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1answer
543 views

Can the opposite of an elementary topos be an elementary topos?

This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose ...
2
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0answers
262 views

Is the category of quasi-coherent sheaves not a topos?

There are two parts to my question: Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos? My reasoning is thus: It ...
9
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2answers
247 views

Simplicial spaces internally to simplicial sets

I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual). After finding out about some of the uses of simplicial ...
21
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3answers
1k views

Is there a good general definition of “sheaves with values in a category”?

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
5
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1answer
180 views

Hakim's definition of a locally ringed topos

In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied: (i) For ...
3
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0answers
44 views

Compact “subsets” in the internal real numbers of a topos

Let $\mathcal{E}$ be a topos with internal (Dedekind?) real numbers $\mathbb{R}_{\mathcal{E}}$. Suppose that $K\subset\mathbb{R}_{\mathcal{E}}$ is a compact "subset" (here, I assume that ...
10
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650 views

The “unification” of geometry via topos theory?

This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question. There has been quite a lot ...
5
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1answer
266 views

Is there a Grothendieck correspondence for sheaves/stacks?

Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories $$ \mathsf{DFib}(\mathcal{C}) \cong \mathsf{PSh}(\mathcal{C}), $$ whereas the Grothendieck ...
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Meta property criterion on the internal language of a topos of sheaves for Noetherianity

Let $X$ be a topological space. Then $X$ is irreducible if and only if in the internal language of $\mathrm{Sh}(X)$, $\bot$ is not true and we have one side of De Morgan's law, that is, for all ...
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128 views

Big Zariski topos and classifying topos of local rings

In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are ...
6
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2answers
264 views

Characterization of 'canonical' natural numbers objects

Canonicity is a useful property satisfied by some type theories, saying that every element of natural number type is propositional equal to an element of the form $s^n(0)$, where $s$ is the successor ...
12
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1answer
608 views

So after all, what is this thing about topos theory and non-binary truth?

Disclaimer. The question below is necessarily vague. I understand neither the subject matter topos theory nor the object about which my question is (the construction of a fractional / non-binary / ...
8
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1answer
254 views

A weak form of countable choice

Let $\Omega$ be the set/type of truth values. We're using constructive logic. Define $AC_{0, 0} = \forall P : \mathbb{N}^2 \to \Omega, (\forall n \in \mathbb{N}, \exists m \in \mathbb{N}, P(n, m)) \to ...
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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 ...
3
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1answer
261 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,...
23
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1answer
656 views

Is the opposite category of commutative von Neumann algebras a topos?

By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict ...
4
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1answer
223 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 (=...
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242 views

Describing the points of a glued topos

Let $f : \mathbf{X}\to \mathbf{Y}$ be a morphism of topoi; in his 1977 monograph, Johnstone describes the open mapping cylinder of $f$ as the following pushout of topoi: $\require{AMScd}$ \begin{CD} \...
5
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1answer
382 views

About an argument in Olsson's book

The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson. I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end. It seems that there might ...
12
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1answer
243 views

Stone duality for the algebra of Boolean functions such that $f(\top,\dots,\top) = \top$, or: What does the presheaf topos on $FinSet_\ast$ classify?

$\newcommand\FinSet{\mathit{FinSet}}\newcommand\FinBool{\mathit{FinBool}}\newcommand\FreeFinBool{\mathit{FreeFinBool}}\newcommand\Set{\mathit{Set}}\newcommand\Psh{\mathit{Psh}}$It's well-known that ...
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Is there a concrete application of topos theory?

The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But ...
17
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1answer
338 views

In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?

For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
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92 views

Is the category of covering spaces always a topos?

It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...
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3answers
699 views

Resources for topos theory

I am trying to learn topos theory and I am finding a strong scarcity of resources. Is there any canonical textbook to refer someone to when learning this topic? So far, I have only been able to find ...
6
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1answer
389 views

Different definitions of condensed sets

The $\kappa$-condensed sets are defined as the sheaves on the site of profinite spaces of cardinality less than $\kappa$ (with $\kappa$ an uncountable strong limit cardinal) with morphisms the ...
8
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1answer
277 views

Is ${\bf Set}$ the terminal autological topos

An autological topos is a type of topos defined by Mike Shulman in his paper on stack semantics; specifically, they are toposes satisfying an additional topos theoretic axiom schema expressed in their ...
10
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1answer
928 views

Computations in condensed mathematics, page 32-34

I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions: At the last line of pg 32 - it seems to imply that ...
10
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0answers
345 views

How do properties of a partial order $\mathbb{P}$ affect the logic of the functor category $\mathsf{Set}^\mathbb{P}$?

$\DeclareMathOperator\true{\mathsf{true}}$I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway. I am ...
13
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4answers
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Two interpretations of implication in categorical logic?

I am a bit confused about the interpretation of "implication" in the standard treatment of categorical logic, for example in [Bart Jacobs 1999] "Categorical Logic and Type Theory". ...
10
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1answer
566 views

When does a topos satisfy the axiom of regularity?

In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-...
9
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1answer
313 views

From Topoi to Grothendieck categories

This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
8
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1answer
268 views

What's the localic reflection of a presheaf topos?

$\newcommand{\Psh}{\operatorname{Psh}} \newcommand{\Sh}{\operatorname{Sh}} \newcommand{\O}{{\mathcal{O}}}$ Let $X$ be a locale, $\O(X)$ the corresponding frame. What's the localic reflection of $\Psh ...
16
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1answer
470 views

Toposes with only preorders of points

For a Grothendieck topos $\mathcal{E}$, are the following assertions equivalent? $(i)$ $\mathcal{E}$ is localic. $(ii)$ The diagonal geometric morphism $\mathcal{E} \to \mathcal{E} \times \mathcal{E}$ ...
8
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1answer
171 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 ...
13
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1answer
430 views

Does every category with a subobject classifier embed into a topos?

I've never seen an example of a category with a subobject classifier which didn't embed nicely into a topos. Is there a good reason for this? Question 1: Let $\mathcal C$ be a category with a ...
3
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1answer
172 views

Alternative definition of power object in a category

The standard definition of a power object seems to be: objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \...
3
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1answer
183 views

Deligne's theorem for $n$-topos

Deligne's theorem states that a coherent topos has enough points, i.e. that we can prove that a morphism of sheaves on a "nice" site is an isomorphism by showing that the induced morphism on ...
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158 views

Subobject classifiers with a quantale structure

Quantales were kind of popular a few decades ago, as some sort of "quantum" version of the much more widespread concept of locale (Mulvey introduced them to study $C^*$ algebras). I was ...
8
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0answers
319 views

Geometric stacks, groupoids and étendues

If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent: Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...
4
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1answer
289 views

Objects and morphisms in inverse limits of toposes?

Certain Galois toposes can be written as $\lim_{i \in I} \mathbf{PSh}(G_i)$ where $(G_i)_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of ...
7
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1answer
103 views

Prove that a Boolean two-valued topos in which supports split is well-pointed

In Lawvere and Rosebrugh's Sets for Mathematics, they write It is a theorem [MM92] that a topos is well-pointed if and only if it is Boolean, two-valued, and supports split. [MM92] is a reference to ...
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188 views

Lawvere's Perugia Notes

A PDF version of Lawvere's Perugia Notes (Theory of Categories over a Base Topos) can be found here: https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf That PDF ...
7
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0answers
144 views

Internal logic of the small étale topos of an algebraic variety

If we consider the internal logic of the small étale topos of an algebraic variety, is the variety's geometry reflected in it? For instance, how different are the internal logics for the étale topoi ...
15
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1answer
869 views

Reconstruct a variety from its crystalline topos

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point. Can we reconstruct $X$ from its small crystalline topos $((X/...
5
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0answers
145 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 ...
13
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2answers
675 views

Can one associate a “nice” topos to a von Neumann algebra?

The question here inspires my present question. Reyes proves here that the contravariant functor Spec from the category of commutative rings to the category of sets cannot be extended to the category ...
2
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1answer
113 views

Constructivity of two problems on a standard simplex?

Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and non-negative and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always ...

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