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6
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1answer
467 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
51 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
45 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
47 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
52 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
70 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
67 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
90 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
93 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
40 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
71 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 ...
10
votes
3answers
266 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
234 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: ...
1
vote
2answers
85 views

Do 'nice' first order logics have universal models?

A first-order logic is interpreted in a model where sentences of the logic can be said to be true or false. There may be more than one model, and we can identify morphisms between models. Do we have ...
8
votes
1answer
361 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
76 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
622 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 ...
2
votes
2answers
81 views

Definition of ordinals in Grothendieck toposes

What is a useful definition of an ordinal in Grothendieck toposes. Useful in this context means that I would like to construct fixed points of monotone maps on complete lattices using iteration from ...
2
votes
1answer
77 views

Projective Objects in a Topos

I'm working a problem in MacLane and Moerdijk, and am not sure how to proceed. The end goal is to show that if $Sh(X)$ has enough projectives and $X$ is $T_1$, then $X$ has a basis of clopen sets. I ...
4
votes
1answer
248 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 ...
7
votes
0answers
161 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 ...
2
votes
1answer
89 views

Does this right adjoint of a geometric morphism preserve directed colimits?

Let $E$ be a sheaf topos $E=\operatorname{Shv}(C)$ and $\{x_i:\operatorname{Sets}\to E\}$ be a set of geometric morphisms $x_i:Sets\to E$ such that the morphism $$ x^*\colon E\to \prod_i ...
4
votes
1answer
66 views

Sites for étale $G$-spaces

This is a problem Sheaves in Geometry and Logic by MacLane and Moerdijk (problem 3.10) Suppose we have a space $X$ on which a discrete group $G$ acts by homeomorphisms. Given an étale space $p:E ...
4
votes
1answer
122 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
89 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
165 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
348 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. ...
1
vote
2answers
74 views

Plus construction of a presheave factors every sheaf-valued morphism.

I'm having some trouble understanding the correctness of some proof in Sheaves in Geometry and Logic (Mac Lane, Moerdijk). It concerns the lemma III.5.3 : If $F$ is a sheaf and $P$ a presheaf, ...
0
votes
1answer
85 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 ...
2
votes
1answer
37 views

Commutativity of a sheaf of groups from an epimorphism

Let $F$ and $G$ be sheaves of groups $\mathcal{S}^{op}\to Groups$ and $f:F\to G$ an epimorphism (of sheaves of sets). If $F$ is a sheaf of commutative groups, is $G$ also a sheaf of commutative ...
4
votes
1answer
181 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 $X$ ...
0
votes
1answer
98 views

Two questions about monomorphisms and pullbacks in a Grothendieck topos

Let $\mathbf{C}$ be a Grothendieck-topos and $f:Y\to X$ a morphism. Is the pullback $W\times_X Y\to Y$ of a split monomorphism $g:W\to X$ along $f$ again a split monomorphism? I don't think this ...
1
vote
0answers
84 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
68 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
357 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
73 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
166 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 ...
6
votes
1answer
114 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 ...
5
votes
0answers
77 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 $f$ ...
4
votes
0answers
72 views

Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
2
votes
1answer
61 views

Joyal-Tierney definition of locally isomorphic objects

I am struggling with Joyal-Tierney's paper Strong stacks and classifying spaces, (appeared in "Category Theory (Como, 1990)", volume 1488 of LNM, pp. 213–236, Springer 1991). In particular one of the ...
3
votes
2answers
66 views

If $\mathcal F$ is a sheaf, then is $\mathcal F (- \times X)$ a sheaf?

Let $\mathcal F$ be a sheaf of sets on a site. Fix an object $X$ of the underlying category of the site, which is assumed to contain a final object and have products. Define a presheaf $\mathcal G$ ...
10
votes
2answers
208 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 ...
3
votes
1answer
188 views

Reference for forcing using topos theory

I've just saw in Maclane and Moerdijik's book ("Sheaves in Geometry and Logic: A First Introduction to Topos Theory") about the Cohen forcing viewed in a categorical way using Topos theory. Is there ...
0
votes
1answer
56 views

Is this thing K-finite?

This is related to this question: Freyd's Geometric Finiteness : An Example Computation I've essentially reduced the problem to the following question: Equip $\mathbb{N}$ with the discrete ...
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 ...
2
votes
1answer
97 views

Is the category of sheaves on a site always abelian?

Let $\mathcal{C}$ be a site and $\mathcal{A}$ be an abelian category. Suppose that the category of presheaves $$ Psh(\mathcal{C},\mathcal{A}) = \operatorname{Fun}(\mathcal{C}^{op},\mathcal{A}) $$ is ...
1
vote
0answers
55 views

A sheaf of cumulative hierarchies

I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on ...
5
votes
2answers
264 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 ...