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 is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
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Is there a topos-like category that classifies regular subobjects?

A quasi-topos is a category that characterised by being finitely complete and finitely cocomplete that is also locally cartesian closed and has a strong sub-object classifier. A topos is finitely ...
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Elementary topoi have initial objects, why?

An elementary topos is a category that: has finite limits is cartesian closed has a subobject classifier and one can show (with quite a bit of effort) that it has finite colimits. Is there, ...
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Different definitions of cartesian closed category?

The following is the definition of a cartesian closed category in Goldblatt's Topoi: Definition 1: A category $C$ is cartesian closed if (1) it is finitely complete, i.e. every finite diagram ...
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Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to $\...
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Two questions on completely regular filters in locales

I'm reading the exposition of the Stone-Čech compactification for locales in Johnstone's book Stone Spaces. In Chapter IV Paragraph 2.2, Johnstone constructs the Stone-Čech compactification of a ...
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Where to study $2$-category theory?

Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the ...
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68 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
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“External” Lawvere-Tierney Topologies?

Suppose I have a map $j : \text{Sub}(1) \to \text{Sub}(1)$ from subterminal objects of a topos to themselves which satisfies analogous axioms to those of a Lawvere-Tierney topology, namely $j(1) = 1$, ...
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What are the universally effective epimorphisms of topological spaces?

An effective epimorphism in a category is a morphism that is the coequaliser of its kernel pair, and a universally effective epimorphism is a morphism $f : X \to Y$ such that, for every pullback ...
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Universal Local Ring Object in the Zariski Topos

I am trying to understand the universal local ring object in the Zariski topos as explained in Sheaves and Geometry and Logic by Mac Lane and Moerdijk (page 451); but I am unable to penetrate the ...
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When the empty family of arrows to an object is epimorphic, that object must be initial?

Is it true that when the empty family of arrows to an object $E$ in some category is epimorphic, that object $E$ must be the initial object $0$? This is a claim on page 433 (eq. 22) of Mac Lane and ...
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Are intersections of essential sublocales essential?

A sublocale $X_j$ of a locale $X$, given by a nucleus $j : \mathcal{O}(X) \to \mathcal{O}(X)$, is called essential (sometimes also principal) if and only if the following equivalent conditions are ...
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Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and $...
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Concrete description of (co)limits in elementary toposes via internal language?

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any ...
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Converse to pullback pasting and local diffeomorphisms

The nlab page on local diffeomorphisms gives the following two equivalent conditions for a smoonth function to be a local diffeomorphism. The derivative is an isomorphism of tangent spaces $df:T_xX\...
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45 views

Toposes in algebraic geometry

I know the definition of topos and read an introduction to algebraic geometry. I heard that topos is used for algebraic geometry and want to know detail. Where can I read about this?
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Do 'nice' first order logics have universal models?

A first-order logic is interpreted in a model where sentences of the logic can be said to be true or false. There may be more than one model, and we can identify morphisms between models. Do we have ...
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question about a theorem in Maclane-Moerdijk's “Sheaves in Geometry and Logic”

Like this questioner I am trying to understand the proof of Theorem 2 of Section 5, Chapter I, of MacLane-Moerdijk's "Sheaves in Geometry and Logic". I am wondering, how do you define the functor $L$...
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Extra hypotheses in proposition in Sheaves in Geometry and Logic?

In chapter I, section 9, proposition 5 in Mac Lane and Moerdijk's Sheaves in Geometry and Logic, it is stated that if $f : B' \to B$ is a morphism in a complete category $\mathcal{C}$ and the category ...
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32 views

Commutation of pullbacks with disjoint unions in sites?

From what I understand, in superextensive sites disjoint unions commute with pullbacks. I know this holds in every topos sense pullback has a right adjoint, and I have heard something along the lines ...
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Reference for forcing using topos theory

I've just saw in Maclane and Moerdijik's book ("Sheaves in Geometry and Logic: A First Introduction to Topos Theory") about the Cohen forcing viewed in a categorical way using Topos theory. Is there ...
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Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
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Effective equivalence relations in a topos

I have a question about Johnstone's proof (in either Topos Theory or the Elephant; the accounts are essentially the same, so far as I can tell) that internal equivalence relations in a topos are ...
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What is the internal hom functor in the context of an internally projective object?

I am trying to understand the definition of an internally projective object from nLab. It says that an object $E$ of a topos $\mathcal{T}$ is called internally projective if the internal hom functor $...
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Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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Strange definition of spectrum?

The following is taken from these notes. Definition 2.8. Let $\mathscr E$ be some cartesian closed category and let $A$ be an $R$-algebra. The spectrum $\operatorname{Spec}_A(R[x_1,\dots ,x_n]/I)$ of ...
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Does there exist a topos whose terminal object is not a generator, and every injective morphism is monic?

Say that a morphism $f : X \to Y$ in a category is injective if, for each pair of parallel morphisms $x, x^\prime : 1 \rightrightarrows X$ from the terminal object, if $f \circ x = f \circ x^\prime$, ...
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2answers
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subobject classifier for partial orders

Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.) If not, what is a category which is "close" to the category of partial orders (e.g. it should consists ...
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63 views

Is there a reasonable Grothendieck topology on the category of modules over a ring?

How about over a field (i.e. f.d vector spaces)? Can these categories be considered as a site?
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What does it take to divide by $2$?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That's because the axiom of choice allows for the definition of ...
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Could there be an “$n$-th root” of the category $\mathsf{Set}$?

Here is a thought experiment: Suppose we did not know what sets and functions are. The general idea of a topos is, that it somehow serves as a foundation for mathematics. So let there be an alternate ...
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Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
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Sheaves in Grothendieck Topologies

Let $S$ be a scheme and the category of $S$-schemes be equipped with one of the standard Grothendieck topologies, say étale or fppf. Let $G \rightarrow H$ be a morphism of abelian sheaves on this ...
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Examples of toposes in which the Axiom of Determinacy holds.

I just stumbled upon the Axiom of Determinacy which is an axiom in set theory - inconsistent with the Axiom of Choice, consistent with the Axiom of Dependent Choice, that states that for every subset $...
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Constructive proof that the Lie functor is faithful?

I am wondering how to show that the Lie functor taking Lie groups to Lie algebras is faithful? In particular, I am looking for a constructive proof, since I am working in the context of synthetic ...
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Notation for inhabited sets

A set $X$ is called inhabited if it has some element. In classical mathematics, this means that it is not the empty set, so that one usually writes $X \neq \emptyset$. However, in intuitionistic ...
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1answer
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Internal semantics of a category based on a fixed topos.

I'm not certain as to how I should formulate this question; it might be considered a soft question. I am interested in finding a general way to take a category $\mathbb{C}$ and an (elementary) topos $...
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Is every topos equivalent to a full subtopos of U-small objects in another topos?

Let $\mathcal{E}$ be a topos, and $\mathcal{U}$ an $\mathcal{E}$-universe (as discussed on this nLab page). Let $\mathcal{E}_\mathcal{U}$ be the full subcategory of $\mathcal{U}$-small objects in $\...
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Description of generated Grothendieck topology

Let $C$ be a small category, and let $\tau$ be a set of sieves in $C$. Assume that $\tau$ contains all the maximal sieves, and is stable under pullbacks. How to describe the Grothendieck topology $\...
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How does the internal language of a topos come to be?

There are several books and articles on topos theory which mention the internal language, but I can't manage to see the big picture from any of them. I would like a soft explanation of how the ...
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Can curvature be defined in Topos Theory?

I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context. Topos theory claims that it is a geometric ...
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Covering sieves in a Grothendieck topology

I'm trying to get my head around some of the basics of Grothendieck topologies. Let $(\mathcal{C}, J)$ be a site, let $U$ be an object of $\mathcal{C}$ and let $J(U)$ be the set of covering sieves on ...
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35 views

What does it mean for a topos to be “generated” by some kind of objects?

Is there a universal definition of what that phrase should mean? Suppose we are considering objects of a topos with a particular property, call them $P$-objects (I have in mind the case where P="...
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Stack semantics for the layman

I am trying to read bits and pieces of Ingo Blechschmidt's notes on using the internal language of toposes in algebraic geometry. I have not studied the internal language. I only have a bare bones ...
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Replacement in a topos with an eye to a natural model of TST

TST is a typed first order set theory that is essentially the friendly, simple version of Russell's type theory. There are an infinite number of sorts indexed by the natural numbers, each variable ...
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topos defined by sets

Let $C=Sets$ be the category/site of sets, equipped with the topology defined by surjective families. Why is the associated topos $T$ equivalent to the punctual topos $Sh(pt)\simeq Sets$? (This is ...
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I want to prove that $\text{Sh}(C_{\mathbb{T}},J)$ is the classifing topos for the theory of $\mathbb{T}$-local algebras.

Let $\mathbb{T}$ be a essentially algebraic theory, $C_{\mathbb{T}}$ be its syntactic category and $J$ be a subcanonical coverage on $C_{\mathbb{T}}$ Then, I want to understand why "$\text{Sh}(C_{\...
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Enough injectives in a topos

In the category of sets, each nonempty set is injective since given a mono $A\ \rightarrowtail B$ and an arrow $A\rightarrow C$ we can lift to an arrow $B\rightarrow C$ by giving up injectivity : send ...
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Gerbes and Brauer group

Let $µ$ be a sheaf of abelian groups on a site $C$. There is a bijection between isomorphism (equivalence) classes of µ-gerbes over $C$ and $H^2 (C, µ)$. Can someone give me a good reference for the ...