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10
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3answers
376 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
273 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
91 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
383 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
97 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 ...
30
votes
1answer
879 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
33 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
89 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
103 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 ...
5
votes
1answer
286 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
193 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
110 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
70 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
149 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 ...
5
votes
1answer
138 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 ...
6
votes
1answer
192 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 ...
16
votes
4answers
464 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
87 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
102 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
39 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
250 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
131 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
106 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
82 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
487 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
86 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
183 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
136 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
87 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
77 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
80 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
70 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
259 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 ...
5
votes
1answer
263 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
60 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
106 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
117 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
61 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 ...
7
votes
2answers
291 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 ...
4
votes
1answer
125 views

What does the maximal basis for a Grothendieck topology look like?

There is the notion of Grothendieck topology on a category $\mathbf{C}$. This is a specific assignment of a collection $J(C)$ of sieves to each object $C\in \mathbf{C}$. A Grothendieck pretopology or ...
4
votes
1answer
57 views

Does a natural transformation on sites induce a natural transformation on presheaves?

Suppose $C$ and $D$ are sites and $F$, $G:C\to D$ two functors connected by a natural transformation $\eta_c:F(c)\to G(c)$. Suppose further that two functors $\hat F$, $\hat G:\hat C\to\hat D$ on the ...
5
votes
1answer
467 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 ...
4
votes
1answer
171 views

Why do we need a pullback for the definition or classification of subobjects?

Regarding the subobject classifier construction, why do we need the pullback? Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X ...
7
votes
3answers
231 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 ...
2
votes
1answer
60 views

Example of a functor from the slice topos into the whole topos which does not preserve monomorphisms

For an object $c$ of a site $C$ with terminal object $*$ there is a functor \begin{equation} f:\operatorname{Sh}(C)/c\to \operatorname{Sh}(C)/*=\operatorname{Sh}(C) \end{equation} from the slice topos ...
1
vote
1answer
60 views

Right adjoint to the product in an over category

Let $\mathbb{C}$ be a small category and $X\in \mathbb{C}$ an object. The category $\widehat{\mathbb{C}}$ of presheaves on $\mathbb{C}$ is cartesian closed, i.e. each product \begin{equation} -\times ...
4
votes
0answers
165 views

Where else has Proposition B1.3.17 in the Elephant been proved?

This is a sort of reference request. Proposition B1.3.17 in Johnstone's Elephant reads: Proposition 1.3.17 Let $\mathcal{S}$ and $\mathcal{T}$ be categories with pullbacks, $F \colon \mathcal{S} ...
2
votes
1answer
104 views

Question on the relation between sheaves over an object and sheaves on a category over that object

The Proposition appearing in the wonderful answer of Zhen Lin to this other question states that for a small category $\mathbb{C}$ and an object $X\in \mathbb{C}$ (*)\begin{equation} ...
2
votes
2answers
438 views

What is a Lawvere-Tierney topology?

I've read some articles and books for the definition and use of Lawvere-Tierney topologies, but I still don't understand their role. Some people introduce these topologies as modal operators for ...
11
votes
1answer
213 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 ...