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41 views

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
57 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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0answers
31 views

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
66 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
66 views

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

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
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2answers
141 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
46 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
87 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
88 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
45 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 ...
1
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1answer
35 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
78 views

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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0answers
54 views

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 ...
3
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0answers
34 views

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 ...
3
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3answers
133 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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0answers
96 views

Compare étale morphisms in Top and Schemes

Perhaps because I find the algebraic definition of étale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of étale morphisms of topological spaces. In topological ...
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2answers
126 views

The dense topology

The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z ...
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0answers
48 views

dense generators and left Kan extensions

Here is an exercise from Borceux, Handbook of Categorical Algebra I, p 174: Consider a category $\mathfrak{C}$, a family $(G_i)_{i \in I}$ of objects of $\mathfrak{C}$ and the corresponding full ...
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1answer
72 views

Are there any non-trivial finite elementary topoi?

Title basically says it all: are there any finite topoi (that is, finite set of objects, finite hom-objects) other than $\textbf{1}$ (the terminal category) and $\textbf{2}$ (the category $\ast ...
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0answers
56 views

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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0answers
59 views

Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
4
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1answer
59 views

Quantificators vs pullbacks

Let $\mathscr C$ be a cartesian category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$. Consider the following pullback square: $\require{AMScd} \begin{CD} P @>{\bar ...
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4answers
66 views

How do you work with infinitesimal exponents in synthetic differential geometry?

I just read this paper by Andrej Bauer, which discusses the basic tenets of synthetic differential geometry. Namely, that for any function $f$, any real number $x$, and any infinitesimal $\epsilon$ (a ...
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1answer
25 views

Surjection Vs Surjective geometric morphism

Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is ...
1
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1answer
104 views

Toposes and Stone-type dualities

The duality between the category Sets and the category CABool of complete atomic boolean algebras is an example of a general Stone-type duality. At the same time, Sets is a topos. Are there other ...
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2answers
41 views

Precomposition with a faithful functor

If $F: C \rightarrow D$ is a faithful functor, then is the precomposition with $F$ functor $F^{\star}:[D:\mathbf{Set}] \rightarrow [C:\mathbf{Set}]$ faithful?
2
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2answers
68 views

How do you see that this diagram is a pullback square?

I am trying to understand why the left-hand square of the diagram below (in a topos) is a pullback, where $\Delta_B$ is the diagonal map, $\delta_B$ is clearly the characteristic map of $\Delta _B$ ...
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0answers
84 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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1answer
30 views

Properties of the $[S,Sets]$, where $S$ is small

I am very new to topos theory and am interested in a couple little properties of a certain elementary topos. Suppose $S$ is a small concrete category. Then I was wondering.. which of there ...
2
votes
1answer
67 views

Every Presheaf Is Colimit of Representables

I am working on the proof that each presheaf $F\in \mathbf{Set}^{\mathcal C^{op}}$ is the colimit of $\mathbf{y}\circ \pi\colon \int_{\mathcal{C}}F\to \mathbf{Set}^{\mathcal C^{op}}$ where ...
4
votes
1answer
89 views

Where does one learn the algebraic geometry needed for topos theory?

I am a masters student familiar with category theory. I have started learning topos theory from MacLane-Moerdijk's book "Sheaves in Geometry and Logic: A First Introduction to topos Theory". I get the ...
4
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3answers
79 views

How to look at adjunctions correctly?

I am learning some category theory to help me with my area of research. I am trying to get familiar with the notion of adjunction. In some books I see the authors proving that two functors form an ...
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0answers
48 views

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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2answers
199 views

What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
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1answer
63 views

1-1 correspondence between nuclei and regular monomorphisms of a locale

I am having a little trouble with Theorem 2.3 in Professor Johnstone's book on "Stone Spaces". The theorem depends on a Lemma (which I am not struggling with; I only include it for context) which ...
1
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1answer
39 views

What is the category of internal locales in a topos equivalent to?

I (think) have heard in a conference, in passing, the sentence ''there is an equivalence between internal locales in a topos $\mathbb{S}$ and localic $\mathbb{S}$-topoi''. Is this true in any sense? ...
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1answer
63 views

Coherence isomorphisms in the definition of the descent category.

What does "modulo coherence isomorphisms" mean, in the definition of the descent category of a simplicial topos?
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2answers
65 views

Generators in a topos

What is known about (sets of) generators in an elementary topos ? In particular, does an elementary/Grothendieck topos have a dense set of generators ?
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2answers
118 views

Definition of truth values in a topos.

I'm trying to understand what is meant exactly by a "truth value" in a topos. Take for example the topos of irreflexive graphs. It is known that the classifying morphism can take nodes to 2 different ...
4
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1answer
71 views

Whats the generalisation in category theory of the classical cover in topology?

In topology a cover of a space is a set of subspaces whose union is the space. Obviously a subspace is an inclusion. A space is an object in the category $Top$. In category theory, according to ...
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2answers
48 views

Can the equivalence between principle bundles and maps to classifying spaces be turned into an adjunction.

We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$. This almost looks like an ...
2
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1answer
58 views

Reference for Cech cohomology on sites (not pre-topologies)

I'm searching for a reference dealing Cech cohomology on sites (not pre-topologies). In general, when dealing with Cech cohomology on sites, one admits that the category has finite limits so you can ...
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0answers
99 views

If one knows Homotopy Type Theory, what is the easiest way to learn Higher Topos Theory?

Just read Lurie's book? Do there exist any papers which explain higher topos theory with an audience of homotopy type theorists in mind?
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1answer
62 views

Proof that Beck-Chevalley holds for right adjoints iff it holds for left adjoints

I am looking at Bart Jacob's book "Categorical Logic and Type Theory". The proof of Lemma 1.9.7 is left as an exercise for the reader. It does not seem that easy to me, and i have had quite limited ...
5
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2answers
92 views

Is the map into the terminal object an epimorphism?

Let $C$ be a category with a terminal object $1$. Is the unique arrow from an object into $1$ necessarily an epimorphism? If not, is it an epimorphism if $C$ is a topos?
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0answers
45 views

Why $F^{+}$ is a monopresheaf?

I'm having difficult in proving that given a site $(C, J)$, then $F^{+}(c) = colim_{R \in J(c)} R $ is a monopresheaf, in the sense that there exists at most one lifting of $R \longrightarrow F^{+}$ ...
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1answer
119 views

Properties of the internal language of the category of sheaves.

Consider a simple case of sheaves on some topological space $X$, $\operatorname{Sh}(X)$ (recall that a sieve on $U$ is covering iff its $\operatorname{sup}$ is $U$). All of these are Grothendieck ...
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1answer
154 views

Is there any point in a logician studying $\infty$-categories?

My primary areas of interest lately have been set theory, logic, and category theory, so naturally topos theory has been a large part of what I'm learning (in between getting caught up on some other ...
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1answer
75 views

Example of a coproduct and epi preserving functor $F$ which does not preserve finite colimits

What is an example of a functor $F:T\to S$ between toposes $T$ and $S$ which preserves coproducts and epimorphisms but which does not preserve finite colimits?
5
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1answer
264 views

Volume 3 of Johnstone's “Sketches of an Elephant”

Recently, I read the Chapter 8 of Johnstone's "Topos theory" and got interested in the homotopy and cohomology theory of Grothendieck toposes. So I'm looking for the textbooks expanding these ...