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Textbook for Topos Theory

I'm a Masters student planning to do a thesis on Topos Theory, but I have not decided on which particular aspect of it yet. Most likely I will be dealing with elementary topoi with a very logical ...
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1answer
55 views

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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1answer
75 views

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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1answer
49 views

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 ...
2
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1answer
26 views

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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27 views

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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1answer
70 views

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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1answer
51 views

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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1answer
34 views

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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32 views

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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25 views

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 ...
3
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1answer
118 views

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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1answer
57 views

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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24 views

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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1answer
55 views

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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40 views

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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1answer
42 views

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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1answer
137 views

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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38 views

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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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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1answer
218 views

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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1answer
65 views

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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2answers
96 views

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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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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69 views

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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1answer
1k views

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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1answer
179 views

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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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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1answer
56 views

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 ...
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How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is ...
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1answer
97 views

Is a subobject classifier logically equivalent to set-inclusion?

Can one subsume the notion of set-inclusion $\subseteq$ and $\subset$ with the notion of a subobject classifier, expressed via an injective morphism $\hookrightarrow$? Specifically: Are the ...
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1answer
40 views

Is there a left adjoint to the inclusion of discrete (op)fibrations over $X$ into $\mathbf{Cat}/X$?

This would be intended to be like the adjoint to the inclusion of $Sub(X)$, the subsets of a set $X$ into $ \mathbf{Set}/X $, namely taking the image of a function--except "one level higher".
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1answer
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etale sheafification of a presheaf

Consider the following presheaf on the big etale site of smooth schemes over a field $k$: to every smooth $k$-scheme $U$, associate $$F(U):= \{f: U \to \mathbb A^1_k ~|~ \text{$f$ factors through the ...
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2answers
63 views

Natural numbers object via initial morphism

I assume that a natural number object (or see nLab) can be defined as an initial morphisms. (edit: as in the title, I ment initial morphism, not objects) $\hspace{1cm}$ Thoughts: Probably $X:=1$, ...
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The closure of the graph of a certain composite in a topos.

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$ and let $j\colon \Omega\to\Omega$ be a Lawvere-Tierney topology on it. Assume that, for an object $C$ of $\mathcal{E}$, each ...
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3answers
96 views

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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2answers
80 views

Are there any stable $(\infty,1)$-topoi?

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
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2answers
501 views

What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
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1answer
50 views

how would you define the term “elementary” in the context of categories and sets?

I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I ...
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1answer
96 views

What makes “the topos $\mathbf{M}_2$” such a good counterexample?

I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with ...
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1answer
94 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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1answer
62 views

Strange monomorphism of sheaves

What is an example of a Grothendieck topology $J$ on a small category $\mathcal{C}$ such that the category of sheaves $\mathsf{Sh}(\mathcal{C},J)$ has a monomorphism which is not a monomorphism of ...
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1answer
42 views

Showing $f:a\to b$ is epic iff there exists some iso $g: f(a)\cong b$ with $g\circ f^*=f$.

This is Exercise 5.2.4 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". In any topos, given $f:a\to b$, define $p, q: b\to r$ using the pullback of $f$ along itself like so ...
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1answer
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Verifying a Construction Satisfies the $\Omega$-axiom.

I'm stuck on Exercise 4.5.1 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". It's in the topos $\mathbf{Bn}(I)$ of bundles over a set $I$. Goldblatt asks the reader to verify that $\tag{1}$ ...
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Presheaves in a quasi-topos.

I do believe it is a trivial question. But unfortunately I don't know where I can find an answer. Where could I find the answer to the following question? If $S$ is a small category and $X$ is a ...
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Motivation of Strong Monics

Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where ...
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3answers
166 views

Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
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103 views

Compare étale morphisms in Top and Schemes

Perhaps because I find the algebraic definition of ├ętale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of ├ętale morphisms of topological spaces. In topological ...