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 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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What is Mazzola's “Topos of Music” about?

Disclaimers: I am neither a musician, nor I want to discredit Mazzola's work. Corollary of the first point: please use a plain style, without technical terms in the area of Music Theory. Corollary of ...
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When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
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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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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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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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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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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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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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Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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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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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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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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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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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 ...
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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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Is there a characterization of coverings in subcanonical pretopologies?

Let $\mathcal C$ be a category. A sieve for $\mathcal C$ is called strictly universally epimorphic if it is one of the covering sieves for the canonical topology on $\mathcal C$. SGA4 gives the ...
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Subobject classifier in $Sets^{Q}$

Let Q be the linearly ordered set of rational numbers considered as a category while $R^{+}$ is the set of reals with $\infty$. In $Sets^{Q}$,prove that the subobject classifier $\Omega$ has ...
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Examples of certain types of toposes

I'm looking for examples of (non-degenerate) categories $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^{op}$ are toposes (assuming that such categories even exist). On a related note, ...
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Is Rel a topos?

Is the category Rel of sets and relations a topos? I've done a few Google searches about this question but I haven't found any answers either way. And I can't recall any answers either way in any of ...
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A few questions concerning “universal colimits” and dense generating sets

My questions are triggered by Borceux, Vol 1, Proposition 4.5.6. The relevant part of the book is browsable on Google Books, but i'll go ahead and reproduce at least the statement here anyway: Let ...
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Does internalization loses informations everywhere?

It is well known that a group object in Grp is necessarily abelian. This can be understood as "internalization loses information". Indeed, if one was to study group theory by looking at group objects ...
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Subobject classifier in $Set^{C^{op}}$

I'm reading "Sheaves in geometry and logic" and I'm not sure if i'm understanding some definitions. We have our functor $\Omega$ defined on objects by $\Omega(C)$$=\{$$S|$ $S$ is a sieve on C in ...
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Terminal object in $Set^{C^{op}}$ and subobject classifier.

This is from Sheaves in Geometry and Logic pg 38. I'm not sure if I understood it correctly but the subobject classifier in $Set^{C^{op}}$ when $C$ is a small category is a map (natural ...
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Can a general version of the covariant powerset monad be derived from the universal property of power objects?

As the title asks, I'm wondering if one can generally squeeze a "covariant power object monad" out of a topos (following the usual example in $\mathcal{Set}$ with functor part the direct image ...
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A question about morphisms in a Grothendieck topos

I am not very familiar with topos theory so please excuse me if this is completely trivial. Fix an object $K$ in a Grothendieck topos $\mathcal{G}$. Let $k:0\to K$ be the unique morphism from the ...
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is the property of representability of a sheaf on the big etale site checkable on the small site?

Let $S$ be a scheme and $F$ a sheaf on $(\textbf{Sch}/S)_\text{etale}$, whose restriction to the small etale site $S_\text{etale}$ is representable (in fact in my case this restriction is ...
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When defining a Grothendieck pretopology,can we get away with less than the fibre product axiom?

$\newcommand\restr[2]{{\left.#1\right|_{#2}}}$ I'm fairly new to this whole area, so correct me if there are any technical errors in any of this. The base category for a classical sheaf is the ...
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The terminal object comes for free in the definition of a subobject classifier

This is Fact 1.4 in Tom Leinster's informal introduction to topos theory. It states the following: if there exists a mono $t:T \hookrightarrow \Omega$ that classifies the monos in our category, in the ...
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A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”

I'm reading Gian-Carlo Rota's book "Indiscrete Thoughts". In page 220 I came across a strange quotation with very few explanations: We thought that the generalizations of the notion of space had ...
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Grothendieck Topology on the Category of Elements

We are given a site $(C, J)$ for a small category $C$ and a Grothendieck topology $J$. If $F\in Sh(C, J)$, we take the natural topology $J_F$ on its category of elements $el(F)$ induced by $J$. I am ...
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category of sheaves over an object

I'm looking for a solution of exercice III.8 (b) in Maclane and Moerdijk's book $\textit{Sheaves in Geometry and Logic}$. Where I can find such a solution ? Or can someone describe the topology J' ...
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Revisiting “What is Mazzola's ”Topos of Music“ about?”

This question has been asked here: What is Mazzola's "Topos of Music" about? But I am dissatisfied with the response for several reasons and would like Math SE to revisit this ...
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sheaves of rings and maps to classifying topos

Let $R$ be the category of finitely presented commutative rings (but I don't know how necessary the hypothesis of finite presentation is for my question). Let $Set^R=Fun(R, Set)$ be the category of ...
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About certain regular epimorphisms in a Grothendieck Topos

I am supposed to prove a rather technical property which should hold in any Grothendieck Topos, but I have troubles in accomplishing this task. Here is the context for the question. Let then ...
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Equivalent definitions of regular categories?

maybe this is a stupid question, but I could not solve it after some time of meditation. There are four different notions of regular categories: 1) A cartesian category with coequalizers of kernel ...
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Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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Why are these two definitions of the left-adjoint to $u^p\colon PShv(D)\to PShv(C)$ equivalent?

Suppose $u\colon C\to D$ is a functor between categories. Then there is a functor $$ u^p\colon PShv(D)\to PShv(C) $$ between the associated presheaf categories by precomposition with $u$ as it is ...
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Why is the h-topology not subcanonical?

The h-topology introduced by Voevodsky on the category $Sch/K$ of separated schemes of finite type over a field $K$ is the Grothendieck topology associated with the pretopology whose coverings are of ...