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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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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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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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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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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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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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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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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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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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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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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30 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 ...
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24 views

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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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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23 views

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 ...
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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 ...
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'Smooth' p-adic analysis (perhaps via toposes)

There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there ...
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When does an atomic topos have this property?

If $f:\mathbb{E}\to\mathbf{Set}$ is an atomic (or locally connected if you prefer) Grothendieck topos, when is it the case that the direct image functor $f_*:\mathbb{E}\to \mathbf{Set}$ is faithful?
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Finite Limits, Exponentiation and Sub-Object Classifiers imply Finite Co-Limits

I recently started studying topoi and the book I am using defines them as categories that have all finite limits and co-limits, exponentiation and sub-object classifiers. The book briefly remarks that ...
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What are some topos-theoretic insights about $G$-sets?

Since a $G$-set is just a functor $G\longrightarrow \mathsf{Set}$, the category of $G$-sets seems to be a simple example of a topos. What are some topos-theoretic insights into $G$-sets? Insights ...
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Is the reflector $\mathcal{E} \rightarrow \text{Sep}_j \mathcal{E}$ left exact? (j is a local operator)

According to Mac Lane & Moerdijk, Ch V, Ex 4, the left adjoint $\mathcal{E} \rightarrow \text{Sep}_j\,\mathcal{E}$ of the inclusion is left exact, but the proof is left as an exercise for the ...
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Technical meaning of “profinite circle”

In a private exchange with a professional mathematician, I found the following statement: the "small etale topos" of a finite field is a "profinite circle", and thus looks like circle. Could anyone ...
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Is there another well-pointed elementary topos satisfying internal choice without natural numbers object?

Is there an elementary topos which Is well-pointed Satisfies the internal axiom of choice Does not have a natural numbers object; and Is not the category of finite sets?
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37 views

subobject classifier and characteristic arrow in presheaf topos

I'm reading the page 54 of this pdf but don't understand the definition of characteristic arrow in $\text{Set}^{\mathbb{C}^{\text{op}}}$. Assume that $F, G \in \text{Set}^{\mathbb{C}^{\text{op}}}$, ...
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70 views

What is the “internal language of a topos”?

What does the sentence "[...] these statements should be interpreted, of course, in the internal language of the topos $\mathcal{E}$" mean, in the context of, say, the definition of a groupoid in ...
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78 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
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38 views

A question on a property of geometric morphisms related to locales.

Is the "localic reflection" of a geometric morphism between topoi the same thing as its "localic part"?
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Is this assignment of the topos of sheaves functorial?

Let $\mathcal{C}$ be a site and for any object $X$ of $\mathcal{C}$ denote by $\text{Sh}(X)$ the category of sheaves on the site $\mathcal{C}/X$. My question is, what can we say about this assignment? ...
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Can I define a site as a category endowed with a pretopology instead of a topology?

If $K$ is a pretopology on a category $\mathcal{C}$ and $J$ the topology it induces, are the Grothendieck toposes $\text{Sh}(\mathcal{C},K)$ and $\text{Sh}(\mathcal{C},J)$ the same in general? As I ...
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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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Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves?

It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero. Is there any ...
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Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
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topos have colimits

Define an (elementary) topos to be a cartesian closed category with all finite limits and subobject classifiers. I'm looking for a proof of the fact that a topos also has all finite colimits. I know ...
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49 views

The Relationship between Separable Functors and Faithful Functors

Consider the adjunction $\mathcal{C} \mathrel{\substack{\mathcal{F}\\\rightleftarrows\\ \mathcal{G}}} \mathcal{D} $ together with unit $\eta: I_{\!_{\mathcal{C}}} \rightarrow \mathcal{G} \mathcal{F} $ ...
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Simple question in “Sheaves in geometry and logic”

There's an argument I don't understand in "Sheaves in geometry and logic" by Mac Lane and Moerdijk, that seems a priori easy but I can't see it. Page 174, diagram (10) (involving the power ...
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base change of an equivalence relation of fppf sheaves

Let $S$ be a scheme, $R,U$ be $S$-schemes and $s,t : R \to U \times_S U$ be an equivalence relation i.e. it's a monomorphisme such that for every $S$-scheme $T$, $R(T) \to U(T) \times U(T)$ is and ...
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Fibrations over topoi

Let $\mathcal{S}$ be an elementary topos. What is (exactly) the relation between $\mathcal{S}$-indexed categories and fibrations over $\mathcal{S}$? Where can I read about this? (Or even find the ...
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Relationship between differential cohesion and synthetic differential geometry

I was wondering what is the relationship between differential cohesion and synthetic differential geometry? I know the basics of synthetic differential geometry from Kock's text, but I am not ...
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39 views

Is any reflector from a presheaf category $PSh(K)$ to a topos $C$ necessarily left exact?

Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$. Is $T$ left exact? $D$ ...
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Proving that a category is cartesian closed

Let $Alg(1)$ be a category whose objects are sets with a unary operation, with no axioms. Morphisms of the category are functions of sets which preserve the operation. I need to show that this ...
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Categorical Foundations text

I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a ...
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Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
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How to prove that in this context epimorphisms are 'surjective'

In article AN ELEMENTARY THEORY OF THE CATEGORY OF SETS of William Lawvere I met a proposition left to reader (poor me) and I hope someone can help me with it. It wouldn't surprise me if it is not ...
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Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?

Let $C'$ be the category with objects $C$ and morphism the monic morphisms of $C$. In any topos, $\phi_A: \operatorname{Sub}(A) \cong \operatorname{Hom}_C(A,\Omega)$ and $\phi_A(m) = \chi_m$. This ...
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The definition of the $false$ truth value

In "Topoi: The Categorial Analysis of Logic" by R. Goldblatt the $false: 1 \to \Omega$ truth value is defined as the characteristic arrow of the arrow $0_1: 0 \to 1$. This definition requires that ...
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Can we recover étale, fpqc etc. morphisms of schemes from the affine versions?

$\DeclareMathOperator{\Spec}{Spec}$ Consider the following procedure for defining a class of morphisms of schemes: Take a suitable class of homomorphisms of rings (e.g., canonical maps to ...