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Questions tagged [topos-theory]

A topos (plural topoi, toposes) is a category that behaves like the category of sheaves of sets on a topological space. Topos theory consists of the study of Grothendieck topoi, used in algebraic geometry, and the study of elementary topoi, used in logic.

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What concretely does it mean to "do mathematics in a topos X"?

I don't know if this is the right kind of question, but I've heard people say this phrase "do mathematics in a topos", and the idea that Set is the topos people usually work in, but only one ...
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Is the category of undirected graphs an infinity-pretopos?

Theorem C2.2.8 in "Sketches of an Elephant" by P.T. Johnstone states that an $\infty$-pretopos with a separating set of objects is a topos. However, the category of undirected graphs seem to ...
Necrosovereign's user avatar
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Is every Heyting algebra the subobject classifier of some topos?

Let $\mathcal{E}$ be an elementary topos. I know that the subobject classifier $\Omega$ of the topos is always a Heyting algebra. I’m interested in versions of the converse — given any (let's say ...
safsom's user avatar
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About F.Lawvere article "AN ELEMENTARY THEORY OF THE CATEGORY OF SETS"

reference articles: [L] AN ELEMENTARY THEORY OF THE CATEGORY OF SETS (LONG VERSION) WITH COMMENTARY, F. WILLIAM LAWVERE (http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf) [T] Topos theory, P....
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Show that the smallest such Grothendieck topology is the open cover topology

This is an exercise in Maclane-Moerdijk (III.6.(b)). Let $\mathbf{C}$ be a small category, and $S_\alpha$ any collection of sieves on objects $C_\alpha$. It is easy to show that there is a smallest ...
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What is the motivation of topos theory?

What is the motivation of topos theory? https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k&index=1 So from my understanding, the motivation of a topos is to ...
Sayaman's user avatar
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How should we really think about subobjects in a topos?

Scenesetting Some define a subobject of $X$ as a monic $s \colon S \rightarrowtail X.$ For familiar reasons, though, most officially define a subobject of $X$ as an equivalence class $[s]$ of such ...
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Does an internal topos object of a topos induce an "iterated topos"?

From the internal language of a topos, especially the dependently typed version, it seems reasonable to be able to define the notion of an internal topos object. For example, you would start with an ...
Daniel Schepler's user avatar
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Zorn's Lemma in Topos Theory

I am reading the book "Sheaves in Geometry and Logic". When constructing the topos of sets, there is a proposition (proposition 8 of the "Topos of sets" subchapter) that uses Zorn'...
kabel abel's user avatar
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Can we sheaf-theoretically force a violation of the continuum hypothesis in a (nice) topos which is *bicomplete*?

$\newcommand{\p}{\mathcal{P}}$I recently dug through the exercises and details in Mac Lane and Moerdijk's book "Sheaves in Geometry and Logic" which concern themselves with (a baby version ...
FShrike's user avatar
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Are there Kripke-Joyal semantics for terms which involve arrows of the topos (rather than purely variables and connectives)?

I follow Mac Lane and Moerdijk's (slightly informal) presentation of the Mitchell-Benabou language. There, they outline several recursive rules to describe what the terms of the language are and ...
FShrike's user avatar
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A crystal on $\operatorname{Cris}(X/S)$ is the same as giving a projective system of crystals on $\operatorname{Cris}(X/S_n)$ with compatability?

I'm currently reading Berthelot- Cohomologie cristalline des schémas de caractéristique p>0 and I'm having a question on page 228. He claims that if $X=\operatorname{Spec}(k)$ for a perfect field $...
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A 'negation-complement' in a topos is a pseudo-complement: a more direct proof than Goldblatt's?

Given a subobject $s\colon S \rightarrowtail X$, we define one sort of complement $\overline{s}\colon \overline{S} \rightarrowtail X$ in the usual way by pulling back $\top$ along $\neg\chi_s$ (or ...
Peter Smith's user avatar
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Is the Jónsson-Tarski topos local?

The category of Jónsson-Tarski algebras is a topos. It is presented as such in Example A2.1.11(i) in Johnstone's Sketches of an elephant. A geometric morphism is local if its direct image functor has ...
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Understanding lemma A.1.6.3 of the elephant 🐘

Let $\mathcal C$ be a cartesian category with a subobject classifier $\Omega$. (i) For every object A of $\mathcal C$, the preorder $\text{Sub}(A)$ is cartesian closed; and in fact $\Omega$ has the ...
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Are all unions in a topos with complete subobject lattices secretly colimits? On a logical analogue of the AB5 axiom

To clarify, here “topos” always means an elementary topos; I do not assume sheaves on a site, where I already knew my question to have a positive answer. It is known but not so immediate from the ...
FShrike's user avatar
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If presheaves over C are generated by subterminal objects, is C is a posetal category?

I am trying to understand one of firsts statements in the proof of the lemma C5.2.4 of Johnstone's Sketches of an Elephant. It says that, for a small category $\mathcal{C}$, the existence of a ...
Dylan Facio's user avatar
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What suffices to ensure that, given a natural numbers object $(N, z, s)$, $s \circ z \neq z$?

Define a natural numbers object $(N, z, s)$ for a category $\mathsf{C}$ in the standard way. For any $1 \overset{q}{\longrightarrow} A \overset{f}{\longrightarrow} A$ in $\mathsf{C}$ there is a unique ...
Peter Smith's user avatar
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Understanding the proof of Giraud's Theorem

I am trying to understand the proof of a lemma of Giraud's Theorem in Lurie's notes on categorical logic: https://www.math.ias.edu/~lurie/278xnotes/Lecture10-Giraud.pdf I am confused by the proof of ...
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Name of the direct image functor

For any map, we have the inverse image with two adjoints: the image which is the left adjoint, and the "forall" operator which is the right adjoint. This applies pretty generally. However, ...
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Examples of non-localic topos

A topos is localic if it is a sheaf topos on a locale, or equivalently, every object can be covered by a family of subterminals. The localic toposes classifies the theory of sentences only, so I ...
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Localic slices of an étendue

An étendue is a Grothendieck topos $\mathcal{E}$ containing a well-supported object $A$ such that $\mathcal{E}/A$ is localic. This does not imply that all slices are localic (for example, the topos of ...
Dylan Facio's user avatar
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Monomorphism to $\{\emptyset.\top\}$ implies equivalence to a sentence

I remember from a discussion the statement: In the context category (I think of some geometric theory, but I do not think the property of theory matters.) If $\{\vec{x}. \varphi\}\overset{[\theta]}\to ...
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Explicit geometric machinery to check the truth of FOL formula

It is standard to use the semantics of topological spaces for checking the truth value of propositional formula in the absence of excluded middle. An explicit description is written down here, for ...
Y.X.'s user avatar
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Examples of geometric theory without classical models

I encountered a post on locales and geometric theory here https://grossack.site/2022/05/22/locale-basics.html In about the middle of this blog, the author gives a geometric theory defining a function ...
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Elementary proofs of distributive and other properties of $\land$, $\lor$, $\neg$ in topos?

Take the 'logical' arrows $\wedge \colon \Omega \times \Omega \to \Omega$, $\lor \colon \Omega \times \Omega \to \Omega$, $\neg \colon \Omega \to \Omega$, defined by the usual pullback squares, as in ...
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proposition 6.22 in Awodey's category theory textbook

Proposition 6.22 reads: For every CCC (Cartesian closed category) $C$, there is a poset $I$ and a functor, $$y: C \to \textrm{Sets}^I,$$ that is injective on both objects and arrows and preserves CCC ...
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1 answer
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Zariski opens of an affine space and (po)sites

There is a very neat presentation of the Zariski open sets in $\operatorname{Spec}(R)$: it is freely generated (under arbitrary unions and finite intersections, which distribute over each other) by ...
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0-connective objects in an $\infty$-topos

I am trying to understand Marc Hoyois's proof of the Van Kampen theorem in an $\infty$-topos given as Lemma 6 in : https://ncatlab.org/nlab/files/Hoyois_PlusConstruction.pdf In the course of the proof,...
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Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?

I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
Miviska's user avatar
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Show something is quasi-isomorphism by showing pullback is quasi-isomorphism.

My question is the following: Let $T$ be a topos and $X\to e$ be a covering of the final object $e\in T$ with respect to the canonical topology. We have a morphism of topoi $f\colon T/X\to T$ where $f^...
Enki's user avatar
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Deducing the definition of subobject classifier from Sub functor being representable

As the title says. Let's consider some category $C$ and the contravariant functor $Sub : C^{op} \rightarrow Set$ that for each object gives its set of subobjects. The introductory book by Leinster ...
Julián's user avatar
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Universal effective epimorphisms and jointly epimorphic sieves

I have a question concerning the canonical topology of a topos. By the definition I know, the canonical topology is given by universal effective epimorphisms as the covering sieves. Now I have read ...
Enki's user avatar
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6 votes
1 answer
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Elementary proof that, in a topos, every epimorphism is regular?

Mac Lane and Moerdijk have a neat proof on p. 197 that, in a topos $\mathcal{E}$, every epimorphism $f \colon C \to B$ is a coequalizer. But this depends on the assumption that the slice category $\...
Peter Smith's user avatar
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1 vote
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Are the "intramural" and "extramural" definitions of "set" from category theory equivalent?

Note: This question takes a pluralistic / "multiverse" view of sets (including "classes" and "collections") and set theories. My understanding (from reading many nLab ...
hasManyStupidQuestions's user avatar
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Ordinary functor being E-indexed

I am learning parts of topos theory and it feels like I am missing something regarding indexed functors. This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ ...
Ilk's user avatar
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1 answer
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Doubt in proof showing in topoi, equivalence relations are effective

This is proposition 1.4 (pg. 30) in van Oosten's lecture notes on Topos Theory. Prop: In a topos, every equivalence relation is effective, i.e. a kernel pair. Following is the first half of the ...
Ajin Shaji Jose's user avatar
2 votes
1 answer
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Doubt in van Oosten's Topos Theory notes

I came across this definition while reading Jaap van Oosten's Topos Theory lecture notes (pg 29, def 1.3). In a category with finite limits, an equivalence relation on an object $X$ is a subobject $R$ ...
Ajin Shaji Jose's user avatar
4 votes
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113 views

For which sites are all sheaves connected (as presheaves)?

When $X$ is a topological space, any sheaf $S$ on $X$ is a connected object of the category of presheaves $\widehat{\mathcal{O}(X)}$ in the sense that $\mathrm{Hom}_{\widehat{\mathcal{O}(X)}}(S,A+B) \...
Colin's user avatar
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8 votes
2 answers
436 views

Is axiom of replacement nicely stateable in the language of ETCS?

ETCS has a nice category-theoretic formulation: "well-pointed topos with a natural numbers object and axiom of choice." I'm too new to topoi to really understand all of what's going on, but ...
Cobalt _000's user avatar
1 vote
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72 views

Question regarding the fullness of restriction functor of ${\rm Sh}(X) \to {\rm Sh}(\mathcal{B})$

I am trying to prove that the restriction functor $\mathbb{r}: {\rm Sh}(X) \to {\rm Sh}(\mathcal{B})$ is an equivalence of categories, and I came across this answer (here you can also find the full ...
babu's user avatar
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1 answer
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Question regarding the restriction functor of ${\rm Sh}(X) \to {\rm Sh}(\mathcal{B})$

I am trying to prove that the restriction functor $\mathbb{r}: {\rm Sh}(X) \to {\rm Sh}(\mathcal{B})$ is an equivalence of categories, and I came across this answer (here you can also find the full ...
babu's user avatar
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ETCS direct image functor

In Lawvere's paper on Elementary Theory of Category of Sets (https://artscimedia.case.edu/wp-content/uploads/2013/07/14182623/Lawvere-ETCS.pdf), at page 27, he proves Lemma 3. by constructing a ...
hahaha123's user avatar
4 votes
1 answer
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What is the precise relation between the category Set and the category Grpd?

In nLab we read: Set is the discrete object classifier in the category Grpd of groupoids and functors. But this is far above my head, could you say in more simple terms what is the relation? Set is a ...
user234212323's user avatar
6 votes
1 answer
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A definition for sheaves on stacks

Here's is a definition of sheaves on stacks in the famous Complex Oriented Cohomology Theories and the Language of Stacks (COCTALOS) lecture notes by Mike Hopkins. Since it was written by students and ...
Qi Zhu's user avatar
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4 votes
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$i\times i $ is regular epimorphism when $i$ is in 1.3.5 in Elephant

In 'Scholium 1.3.5' in Sketches of an Elephant, Johnstone claims that if a category has finite limits, reflexive coequalisers and regular epimorphisms are stable under pullback, then it is regular. It ...
Oddly Asymmetric's user avatar
1 vote
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64 views

Substitution lemma for many-sorted logic interpretation in a topos

Goldblatt's "Topoi" suggests the following exercise (16.3.2). Let $\varphi$ be some formula of some many-sorted language, and let $u$ be some term of the same sort as some free variable $v_i$...
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1 vote
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Coarse Object in an arbitrary category

In the Sketches of an Elephant, there is the following notion: We define an object $C$ of a quasitopos $\mathcal{E}$ to be coarse if 'it cannot detect the lack of balance of $\mathcal{E}$; i.e. if, ...
TheWanderer's user avatar
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4 votes
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How to model theories and use internal logic in a topos?

How would I go about modeling theories and doing math in a general topos, instead of just Set. For example, in Set, I might define a groups as "a set which satisfies the following conditions: ...&...
tses's user avatar
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Support of subobject characterized by coproducts of characters is the join of the corresponding subobjects

An exercise in Goldblatt's "Topoi" goes as follows, assuming the existence of set-indexed copowers of 1: Let $\{ a_s \rightarrowtail 1 : s \in S \}$ be a set of subobjects of $1$ in $\...
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