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16
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1answer
184 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 ...
4
votes
1answer
60 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 ...
2
votes
2answers
87 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 ...
3
votes
0answers
41 views

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 ...
4
votes
0answers
58 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 ...
23
votes
0answers
473 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 ...
8
votes
1answer
169 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 ...
5
votes
0answers
73 views

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 ...
1
vote
1answer
55 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 ...
13
votes
0answers
201 views

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 ...
1
vote
1answer
86 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 ...
1
vote
1answer
37 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".
0
votes
1answer
51 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 ...
1
vote
2answers
60 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 $X:=1$, ...
2
votes
0answers
33 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 ...
5
votes
3answers
87 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 ...
0
votes
2answers
76 views

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

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
5
votes
2answers
316 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). ...
0
votes
1answer
49 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 ...
5
votes
1answer
95 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 ...
1
vote
1answer
93 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 ...
1
vote
1answer
53 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
vote
1answer
38 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 ...
3
votes
1answer
83 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 $\tag{1}$ ...
4
votes
0answers
58 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
votes
0answers
41 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
votes
3answers
150 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 ...
0
votes
0answers
99 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 ...
7
votes
2answers
140 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 ...
2
votes
0answers
53 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 ...
5
votes
1answer
86 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 ...
4
votes
0answers
66 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 ...
2
votes
0answers
66 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
votes
1answer
62 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 ...
3
votes
4answers
76 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 ...
0
votes
1answer
33 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
vote
1answer
117 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 ...
2
votes
2answers
44 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
votes
2answers
76 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 $\Delta _B$ ...
4
votes
0answers
86 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 ...
1
vote
1answer
32 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
86 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
117 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
votes
3answers
81 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 ...
3
votes
0answers
56 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 ...
5
votes
2answers
217 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 ...
1
vote
1answer
64 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
vote
1answer
43 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? ...
0
votes
1answer
67 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?
2
votes
2answers
74 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 ?