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23
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1answer
597 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 ...
19
votes
1answer
424 views

How to find exponential objects and subobject classifiers in a given category

In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject ...
14
votes
4answers
341 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. ...
11
votes
1answer
201 views

Does this notion of morphism of noncommutative rings appear in the ring theory literature?

Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
10
votes
1answer
607 views

Fuzzy logic and topos theory

Why doesn't one develop fuzzy logic by extending topos theory, by simply extending the subobject classifier $\Omega$ to the unit interval [0,1]? Have people done that?
10
votes
3answers
252 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 ...
10
votes
2answers
203 views

Natural numbers objects in topoi: Recursion in a parameter

I am currently trying to prove an exercise from Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk about natural numbers objects. First, we have the ...
9
votes
2answers
352 views

Importance of 'smallness' in a category, and functor categories

I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
9
votes
1answer
112 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 ...
8
votes
1answer
357 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 ...
7
votes
1answer
985 views

What is Mazzola's “Topos of Music” about?

Disclaimers: I am neither a musician, nor I want to discredit Mazzola's work. Corollary of the first point: please use a plain style, without technical terms in the area of Music Theory. Corollary of ...
7
votes
3answers
217 views

Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
7
votes
2answers
115 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 ...
7
votes
2answers
333 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 ...
7
votes
2answers
110 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 ...
7
votes
1answer
147 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 ...
7
votes
0answers
158 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
6
votes
2answers
112 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 ...
6
votes
1answer
461 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 ...
6
votes
1answer
232 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: ...
6
votes
1answer
228 views

An example of a Grothendieck topology

A Grothendieck topology on a category $\mathcal{C}$ with finite limits consists of, for each object $U$ in $\mathcal{C}$ a collection $\text{Cov}(U)$ of sets $\{ U_i \to U \}$ such that Isomorphisms ...
6
votes
1answer
113 views

Glueing sheaves on Grothendieck sites

Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf ...
6
votes
1answer
123 views

Does a geometric morphism $f\colon \cal E\to F$ preserves and reflects the subobject classifier?

I'm stuck in the apparently easy exercise in the title; I tried to prove it twice but both arguments were flawed (one of the two: one can easily obtain a natural map $Sub_\mathcal E(A)\to Sub_\mathcal ...
6
votes
1answer
319 views

What do coherent topoi have to do with completeness?

There is a theorem of Deligne that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via geometric ...
6
votes
0answers
137 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
5
votes
3answers
283 views

Example of a small topos

I'm currently trying to understand this article by T. Noll on the topos of triads in music theory (also, this) However, I can't get past section 2.2 where Noll introduces the subobject classifier, ...
5
votes
1answer
70 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 ...
5
votes
1answer
207 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 ...
5
votes
1answer
80 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 ...
5
votes
1answer
160 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 ...
5
votes
1answer
64 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 ...
5
votes
2answers
215 views

Exercise from Leinster's Informal introduction to topos theory

Forgive the basic question (and the typesetting!) I'm a relative novice regarding category theory, but I've recently decided to teach myself at least the rudiments of toposes. Having stumbled upon Tom ...
5
votes
2answers
230 views

Where is the well-pointedness assumption of ETCS used in everyday math?

Where is the well-pointedeness assumption of the Elementary theory of the category of sets (Lawvere's category-theoretic axiomatization of set theory) used in everyday math? Specifically, if you have ...
5
votes
1answer
165 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 ...
5
votes
2answers
262 views

What fragment of ZFC do we need to prove Zorn's lemma?

It is extremely well-known that Zorn's lemma is a theorem of ZFC. My interest is in a certain finitely-axiomatisable fragment of ZFC, sometimes called RZC (restricted Zermelo with choice) or ZBQC. The ...
5
votes
1answer
412 views

What are the algebras of the double powerset monad?

Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image ...
5
votes
0answers
74 views

Essential geometric morphism seen topologically

I know that any geometric morphism between toposes of sheaves on spaces $f^*\colon Sh(X)\leftrightarrows Sh(Y)\colon f_*$ comes from a continuous map $f\colon X\to Y$. But what does it mean for $f$ ...
5
votes
0answers
98 views

Freyd's Geometric Finiteness : An Example Computation

In his paper "Numerology in Topoi" available here: http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html Peter Freyd defines an object $A$ in a topos $\mathcal{E}$ to be geometrically finite if ...
5
votes
0answers
200 views

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of ...
4
votes
2answers
81 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?
4
votes
2answers
177 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 ...
4
votes
3answers
75 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 ...
4
votes
1answer
68 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 ...
4
votes
1answer
87 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 ...
4
votes
1answer
232 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
65 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
1answer
174 views

Pullback of sheaves and pullback of schemes

Let $\mathbb{G}_m$ the multiplicative group, with coordinate ring $\mathbb{C}[x^{\pm 1}]$, and considered as a sheaf of abelian groups over $\mathrm{Spec}\,\mathbb{C}$ in the Zariski topology. Let $X$ ...
4
votes
1answer
53 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 ...
4
votes
1answer
121 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
404 views

Logic and geometry

By delving into topos theory and sheaves one will eventually discover a "deep connection" between logic and geometry, two fields, which are superficially rather unrelated. But what if I have not the ...