1
vote
2answers
31 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
53 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 $\Delta _B$ ...
1
vote
1answer
28 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
48 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
53 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
71 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 ...
2
votes
0answers
37 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 ...
1
vote
1answer
53 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
31 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
57 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
48 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
88 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
61 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
37 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 ...
1
vote
1answer
52 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 ...
4
votes
2answers
74 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?
7
votes
1answer
126 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
54 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
142 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
23 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
100 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
37 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
90 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 ...
5
votes
1answer
425 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
44 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
41 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
45 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
49 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
63 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
57 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
81 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
83 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 ...
1
vote
1answer
66 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 ...
8
votes
3answers
197 views

Is there a category whose internal logic is paraconsistent?

The internal language of topoi is higher-order typed intuitionistic logic. Now according to wikipedia, the dual of intuitionistic logic, in some sense is paraconsistent. They say Intuitionistic ...
6
votes
1answer
213 views

What was the Lawveres explanation of adjoint functors in terms of Hegelian Philosophy?

I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there. According to the wikipedia entry on Categorical Logic: ...
8
votes
1answer
344 views

Examples of mathematical statements made with adjoint functors

I am wondering if it is possible to use the adjoint functors in topos theory for statements in analysis. Any examples would be warmly welcomed. Though I would prefer simpler, atomic, lemmas or ...
1
vote
0answers
64 views

Can curvature be defined in Topos Theory?

I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context. Topos theory claims that it is a geometric ...
23
votes
1answer
497 views

Does “cheap nonstandard analysis” take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the ...
2
votes
0answers
26 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
4
votes
1answer
215 views

Showing that the sheaf-functor $\epsilon: \tilde{\sf C} \to \tilde{\tilde{\sf C}}$ is an equivalence

Let $(\mathsf C,J)$ be a site. Then we have the category of sheaves $\tilde{\mathsf C}$ and the category $\tilde{\tilde{\sf C}}$ of sheaves over $\tilde{\sf C}$ (both considered with the canonical ...
4
votes
1answer
113 views

Is the internal language of a topos complete, sound and effective?

The internal language of a topos is higher order intuitionistic typed logic. Now according to this article in wikipedia higher order classical logic with full semantics is never complete, sound or ...
4
votes
1answer
71 views

Presheaves over sieves and posets

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found ...
5
votes
1answer
140 views

Grothendieck topology on pre/sheaves

Given a Grothendieck topology $T$, say subcanonical, on a category $C$, we are able to talk about sheaves in $(C, T)$. Since pre/sheaves can be viewed as generalized objects of $C$ (via Yoneda ...
14
votes
4answers
328 views

What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. ...
0
votes
1answer
79 views

Commuting square of functors

Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective ...
1
vote
0answers
67 views

Plus construction of sheafification as a colimit of presheaves.

In Sheaves in Geometry and Logic, Moerdijk and Mac Lane construct the associated sheaf on a site $(\mathcal C, J)$ of a presheaf $P$ as $$ a(P) = (P^+)^+ ,$$ where $P^+$ is defined pointwise as ...
5
votes
1answer
61 views

What functors preserve subobject classifiers?

My question is exactly that of the title. Given a topos $T$, is there a natural sufficient condition for a functor $f$ from $T$ to another topos, $S$, to the subobject classifier? (By this I mean of ...
7
votes
2answers
294 views

A locally constant sheaf on a locally connected space is a covering space; Proof?

As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
1
vote
0answers
69 views

Equivalence of groupoids induced by a G-torsor

I'm struggling again with Joyal-Tierney's "Strong stacks & classifying spaces", and in the proof of the main theorem of the paper, where stacks are characterized as internal groupoids $\mathbb G$ ...
5
votes
1answer
159 views

Elephant: how do I prove Lemma 2.1.7, section C2.1?

I'm referring (also for notations and terminology) to P. Johnstone, Sketches of an Elephant. A Topos Theory Compendium. Volume I. Clarendon Press. Oxford, 2002. The Lemma can be found at page 540. I ...