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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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} \...
Jonathan Sterling's user avatar
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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 ...
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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 ...
Tim Campion's user avatar
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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 ...
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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 ...
Gro-Tsen's user avatar
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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\...
Chetan Vuppulury's user avatar
13 votes
3 answers
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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 ...
Sofía Marlasca Aparicio's user avatar
7 votes
1 answer
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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 ...
Steven Gubkin's user avatar
10 votes
1 answer
439 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 ...
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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 ...
Bryan Shih's user avatar
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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 ...
Neil Barton's user avatar
13 votes
4 answers
1k views

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". ...
YKY's user avatar
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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-...
Jordan Mitchell Barrett's user avatar
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1 answer
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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-...
Ivan Di Liberti's user avatar
8 votes
1 answer
332 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 ...
seldon's user avatar
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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 $\mathcal{E} \to \mathcal{E} \times \mathcal{E}$ ...
Matthias Hutzler's user avatar
9 votes
1 answer
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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 ...
Giulio Lo Monaco's user avatar
16 votes
1 answer
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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 ...
Tim Campion's user avatar
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3 votes
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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 \...
Jordan Mitchell Barrett's user avatar
4 votes
1 answer
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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 ...
curious math guy's user avatar
10 votes
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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 ...
Mirco A. Mannucci's user avatar
9 votes
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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 $\tau$-...
Damien Robert's user avatar
4 votes
1 answer
401 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 ...
Jens Hemelaer's user avatar
7 votes
1 answer
181 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 ...
Robin Adams's user avatar
7 votes
0 answers
197 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 ...
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16 votes
1 answer
970 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/...
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238 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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15 votes
2 answers
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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 ...
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2 votes
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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 ...
VS.'s user avatar
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15 votes
2 answers
2k views

Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics ...
dohmatob's user avatar
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11 votes
1 answer
934 views

Why do elementary topoi have pullbacks?

In the book of Szabo "Algebra of Proofs", Definition 13.1.9 introduces an elementary topos as a cartesian closed category with a subobject classifier. On the other hand, many other sources including ...
Taras Banakh's user avatar
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6 votes
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Examples of Heyting categories that are not toposes?

When explaining how Heyting categories can model first order logic it would be nice to be able to give some small example and contrast it with Set-semantics. I realized however that I don't know of ...
rosensymmetri's user avatar
17 votes
1 answer
400 views

Topos extensions

In set theory, starting from a model $V$ of $ZFC$, a forcing notion $\mathbb{P}$, and a generic filter $G \subset \mathbb{P}$ over $V$, we can find a generic extension which is a model of $ZFC$ and is ...
Mohammad Golshani's user avatar
5 votes
0 answers
159 views

Classifying toposes of theories of rings that aren't local rings

The standard uses of toposes in algebraic geometry come from sites that look roughly like the syntactic sites of theories of local rings that they classify. This isn't particularly surprising, since ...
Cameron Zwarich's user avatar
10 votes
1 answer
559 views

Toposes in which countable choice is true but dependent choice isn't

I'd like examples of toposes in which Countable Choice is true but Dependent Choice isn't. I'd prefer examples without Excluded Middle. It's hard to find a natural example.
wlad's user avatar
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7 votes
2 answers
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Explicit description of exponentials of étalé spaces

It is well known that the category $\mathit{Sh}(X)$ of sheaves of sets on a topological space $ X $ is a topos. On the other hand, there exists a natural equivalence of categories between $\mathit{Sh}(...
ARA's user avatar
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Making area/volume calculations that use SIA rigorous

There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples: A proof that $\sin'(0) = 1$. A proof that the surface area of a cone is ...
wlad's user avatar
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27 votes
1 answer
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Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
Emily's user avatar
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9 votes
1 answer
272 views

Tight apartness relations in toposes

A tight apartness relation on a set is a binary relation $\#$ such that the following conditions hold: $x = y$ if and only if $\neg (x \# y)$. If $x \# y$, then $y \# x$. If $x \# z$, then either $x \...
Valery Isaev's user avatar
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16 votes
0 answers
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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 ...
Tim Campion's user avatar
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2 votes
1 answer
360 views

Questions about Geometric Morphisms

I have several questions about geometric morphisms of topoi. It was recommended that I move my question here from Math.Stackexchange, since it would be good to get an expert on topos theory to answer. ...
Ronald J. Zallman's user avatar
15 votes
1 answer
469 views

Do topos-valued sheaves form a topos?

Let $\bf C$ be a category, $\mathcal S$ an (elementary) topos. If $\mathcal S$ is a presheaf category over $\bf D$, then it's easy to see $[\mathbf C^{\rm op},\, \mathcal{S}] \cong [(\mathbf C \times \...
seldon's user avatar
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9 votes
0 answers
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When is the étale topos of a fibre product the fibre product of étale toposes?

In what follows, all schemes are qcqs. Also, let $\operatorname{\acute{E}t}(X)$ denote the petit étale topos of a scheme $X$. Let $Y\to X$ be an $X$-scheme. Say that $Y$ is a special $X$-scheme if ...
Harry Gindi's user avatar
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4 votes
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Sheaf classifier for topoi

Categories of sheaves in $[C, \text{Set}]_{\text{Cat}}$ (functor category) are equivalently left exact reflective subcategories of presheaf toposes. Categories of sheaves on a topos $[C, \text{Set}]_{\...
Ronald J. Zallman's user avatar
8 votes
1 answer
374 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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2 votes
0 answers
194 views

Only discrete topology gives trivial topos?

Given a Grothendieck site $\mathsf{(C,\tau)}$, if the associated Grothendieck topos $\mathsf{Shv(C,\tau)}$ is trivial, i.e. $\mathsf{Shv(C,\tau)}$ consists of the terminal sheaf $*$ only, can we ...
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On provability of false statements in constructive mathematics [closed]

Lagarias "elementary" reformulation of Robin's theorem is that $$\mathrm{RH}\iff\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holds for every $n\geq 1$, where $\sigma(n)$ is the sum of divisors function and $...
VS.'s user avatar
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13 votes
0 answers
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The Barr-Boole-Galois topos; a modification of sets to play well with schemes

William Lawvere, in his 2015 CT talk "Alexander Grothendieck & the Concept of Space", introduced the "Barr-Boole-Galois topos", which plays the role that sets do for topological spaces (with ...
Ronald J. Zallman's user avatar
10 votes
0 answers
372 views

Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...
Neuromath's user avatar
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0 answers
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What does a partial map classifier look like as a sheaf?

[Cross-posted from M.SE, where it didn't get an answer] In constructive logic, it's possible for a set $X$ to satisfy $$\forall x,y \in X.\, x = y$$ while being non-trivial. Such a set is called a ...
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