Questions tagged [topos-theory]

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10
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1answer
668 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 ...
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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 ...
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1answer
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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 ...
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Results of geometric model theory

I read in some places that model theory has useful applications in algebraic geometry. Could someone give me some results that come from applying model theory to algebraic geometry? I also saw that ...
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109 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 ...
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1answer
430 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.
5
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1answer
179 views

Explicit description of exponentials of etale 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}(...
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115 views

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 ...
21
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1answer
749 views

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 ...
6
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1answer
173 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 \...
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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 ...
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1answer
196 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. ...
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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 $[\bf C^{\rm op},\, \mathcal S] \cong [(\bf C \times \bf D)^{\...
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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 ...
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141 views

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}]_{\...
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1answer
234 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 ...
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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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1answer
454 views

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 $...
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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 ...
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130 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 ...
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67 views

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 ...
3
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1answer
252 views

Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos

Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$. Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\...
4
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1answer
211 views

A list of locally finitely presentable topoi that are not coherent

Coherent topoi play an important role in topos theory, especially in the interaction with logic. Their most handy characterization is provided by Johnstone. Sketches, D3.3.1. Every coherent topos is ...
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174 views

An axiom that shows that the real numbers are weakly countable?

Is there a model of Intuitionistic Higher-Order Logic in which the following axiom is true? Covering Axiom: Any true statement of the form $\forall x \in A, \exists y \in B, \phi(x,y)$ gives rise to ...
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1answer
271 views

Free models of finitely presented essentially algebraic theories in elementary toposes?

The following result is well-known in folklore (I think), but I’ve been unable to find a reference in the literature: Let $T$ be a finitely presented essentially algebraic theory, and $\newcommand{\...
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1answer
474 views

Constructing computable synthetic differential geometry?

I'm a computer scientist, not a mathematician, so apologies if I've messed up a lot of things greatly. I've been reading about synthetic differential geometry, and trying to formalize it in Coq. ...
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1answer
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Surmounting set-theoretical difficulties in algebraic geometry

The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
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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 ...
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239 views

Non-linear Galois descent

This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...
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0answers
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Is there an internal notion of “flat continuous presheaf”?

I have asked a version of this question here, but have been unable to receive an answer or devise one of my own, and am venturing that it may be appropriate for Math Overflow. The "internal form" of ...
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2answers
424 views

'Continuity' of the étale topos

In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative ...
14
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1answer
584 views

Has this “backwards” perspective on toposes been studied?

Topos theory can be seen as a categorification of topology via the following analogies. \begin{array}{|c|c|} \hline \text{locales}&\text{Grothendieck toposes}\\\hline \text{open sets}&\text{...
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1answer
147 views

Geometric morphism whose counit is epic

Is there a name for the class of geometric morphisms whose counits are epic (equivalently, whose direct images are faithful)? This notably includes the class of inclusions/embeddings of toposes. By ...
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0answers
126 views

Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?

When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
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357 views

What are the uses of topoi in algebraic geometry today?

Since Grothendieck introduced them in the 1960s, topoi have found many applications in algebraic geometry, category theory, and logic. For instance, they appear in the development of étale and ...
5
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1answer
141 views

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 ...
5
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1answer
367 views

Failure of SVC in Grothendieck toposes

The axiom SVC (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a choice set $A$ such that $X$ is a subquotient of (i.e. admits a surjection from ...
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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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2answers
425 views

Examples of topos 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 ...
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1answer
204 views

A list of proofs of “Coherent topoi have enough points”

For my research I would like to read all the known proofs of the very classical result "Coherent topoi have enough points", by Deligne. Ref 1: D3.3.13 in Sketches of an Elephant provides a very ...
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What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?

Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
3
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1answer
253 views

Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
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2answers
540 views

O. Leroy's thesis on fundamental groupoids

Does someone have a copy of the O. Leroy's thesis: Groupoïde fondamental et théorème de van Kampen en théorie des topos or has the ability to make a digitalization ? The thesis was done at ...
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1answer
263 views

Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?

In the case of 1-categories, we know there is a functor category $PSh(C):=[C^{op},Set]$, where $C$ is a small category, and this functor category is a topos. I am hoping this will extend to the case ...
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96 views

Can the effective topos be seen as symmetric monoidal?

In Example(s) of monoidal symmetric closed category with NNO without infinite coproducts? user Zhen Lin states the effective topos is locally cartesian closed. On nLab we have that locally ...
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Topos over an Accessible and Cocomplete category

Some facts have been drawn to my attention that raise the following two conjectures to my mind: Many good properties of domains, dcpos, are captured categorically by accessible, cocomplete categories....
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1answer
276 views

Reference request about “internal language of categories”

I've tried to become familiar with the so-called "internal language of a category" for the last months. However, I'm still not confident enough when, for instance, I find a subobject (of a given ...
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0answers
195 views

Is every Grothendieck category with a generator a category of sheaves?

The Gabriel-Popescu theorem tells us that every Grothendieck category with a generator is a left exact localization of a module category. I'm interested in a slightly different way of "representing" ...
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0answers
170 views

Simple description of a Grothendieck topology on the opposite of f.p. complex algebras

Let ${\cal A}$ be the category of finitely presented $\mathbb{C}$-algebras. Let $J$ be the largest subcanonical Grothendieck topology on ${{\cal A}^{op}}$ such that the local algebras in $\cal A$ are ...
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Historically, how were Grothendieck topoi motivated?

The question is about how did the person who invented Grothendieck topoi (presumably Grothendieck) arrive at the necessity of a such a notion. I do not know much about the history of the subject. What ...

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