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7
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2answers
151 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
63 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
123 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 ...
6
votes
1answer
93 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
74 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
64 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
83 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
34 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
119 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
89 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
88 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
103 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
152 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
82 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
60 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
228 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
68 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
47 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
68 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
82 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 ?
7
votes
2answers
151 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
votes
1answer
79 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 ...
0
votes
2answers
100 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
votes
1answer
67 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 ...
3
votes
0answers
121 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?
1
vote
1answer
77 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 ...
6
votes
2answers
109 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?
2
votes
0answers
49 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 $R \longrightarrow F^{+}$ ...
9
votes
1answer
132 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 ...
7
votes
1answer
175 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 ...
1
vote
1answer
89 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
votes
1answer
408 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 ...
2
votes
1answer
27 views

Question on connected and locally connected geometric morphisms.

Is the property of a geometric morphism being both connected and locally connected (in the sense of Part C of the Elephant) stable under pullbacks? I know local connectedness is, I am not sure about ...
6
votes
2answers
153 views

ETCS set theory: Are empty sets isomorphic?

Just a quick question about ETCS: Are any two empty sets isomorphic? Here, a set $X$ is empty if there exists no $x \in X$, i.e. no functions $x: 1 \to X$. The reason I'm asking is that I need this ...
2
votes
1answer
86 views

Is the subobject classifier of the sheaves the sheaffication of the one from the presheaves?

maybe this is an idiot question, but I could not solve it. Let $\Omega$ be the suboject classifier in the category $\mathbf{PSh}(X, J)$ where $(X, J)$ is a site, I know that $\Omega(U) \cong Nat(h_U, ...
2
votes
1answer
41 views

Question about Lemma D1.4.4(iii) in the Elephant - possible typo?

Given a morphism $[\theta] \colon \lbrace \bar{x}.\phi \rbrace \rightarrow \lbrace \bar{y}. \psi \rbrace$ in the syntactic category $\mathcal{C}_{\mathbb{T}}$ of a (cartesian) theory, we are told, in ...
4
votes
0answers
110 views

W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved. I would like to know whether the ...
6
votes
1answer
528 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
0
votes
1answer
64 views

Transposition of arrows in toposes (or cartesian closed categories)

Suppose $\mathcal{E}$ is an elementary topos (take as a definition that of Mac Lane and Moerdijk "Sheaves in Geometry and Logic"). I have a problem with a fact concerning the cartesian closure of ...
2
votes
1answer
48 views

Elementary question about subpresheaves

I have encountered the following phrase which I do not understand. The letter $L$ is used for a locale but I guess (?) this holds for other categories as well. The terminal sheaf $1$ is such that ...
0
votes
1answer
55 views

Lifting of a comonad to the presheaf category

Suppose I have a comonad $G$ on a category $\mathbb{C}$. If $C$ is a preorder then I can define a comonad (i.e. interior operator) $\square$ on the set $\mathcal{P}\uparrow(C)$ of upwards closed ...
0
votes
1answer
58 views

Questions regarding a particular category with functors

I have two questions regarding categories and natural transformations. The first question is as follows. Say we have categories $A$, $B_1$ and $B_2$, with functors $F: A \to B_1$, $G: B_1 \to B_2$ as ...
0
votes
0answers
78 views

explicit proof of pullback stability of epics in a topos

As in Explicit construction of a initial object in a topos I'm looking an elementary proof of the fact that, in a topos, epimorphisms are stable under pullback or, equivalently, that images are ...
2
votes
2answers
96 views

Characterization of epimorphisms of sheaves on a site

I'm stuck with a detail in the proof of the characterization of epimorphism of sheaves on a site in the Mac Lane & Moerdijk book "Sheaves in Geometry and Logic". I want to prove that: "If ...
1
vote
2answers
128 views

Explicit construction of a initial object in a topos

Let $\mathcal E$ be a topos as in Mac Lane and Moerdijk. A initial object in $\mathcal E$ can be obtained as the domain of the equalizer of the morphisms $P!,\epsilon P1:P1\to P^31$, where $1$ is a ...
2
votes
0answers
111 views

Is there a topos in which the natural numbers object are the finite dimensional vector spaces?

I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions. Now vector spaces are prominently characterised by dimensionality. This prompted the ...
2
votes
2answers
49 views

Is the cartesian product of objects in an elementary topos cancellative?

My question is the internalization of this question to an elementary topos $C$. Is it true that: For objects $X,Y$ and $Z$ in an elementary topos $C$ with $X\times Y\cong X\times Z$, then also ...
1
vote
1answer
83 views

What is an object classifier and how does give a natural numbers object?

According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object. What is an ...