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5
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1answer
172 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
125 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
80 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
75 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
66 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
229 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
208 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
57 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
101 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
106 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
56 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
276 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
116 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 ...
3
votes
1answer
56 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 ...
4
votes
1answer
428 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
161 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
226 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
53 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
59 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
160 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} ...
1
vote
1answer
96 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} ...
1
vote
2answers
367 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
204 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 ...
0
votes
1answer
236 views

Sheaves in Grothendieck Topologies

Let $S$ be a scheme and the category of $S$-schemes be equipped with one of the standard Grothendieck topologies, say étale or fppf. Let $G \rightarrow H$ be a morphism of abelian sheaves on this ...
0
votes
1answer
121 views

How to verify sheaf condition in this example?

I am learning about sheaves of sets on a site with a subcanonical topology and have a question. $f:A\rightarrow Hom(-,X)$ is a map from a pre-sheaf $A$ (for which I want to verify sheaf condition) ...
1
vote
1answer
111 views

Composing covers with epis

I am beginner of sheaf-theory and beg your pardon for this maybe silly question. Let $\mathcal{C}$ be a Grothendieck site and $T$ the category of sheaves on $\mathcal{C}$ and let $f:X\rightarrow Y$ ...
0
votes
1answer
143 views

Counterexample for a pullback-pushout situation

Suppose you are in the category of sets or more generally in a topos (i.e. sheaf topos) and $f:A\rightarrow C$, $g:B\rightarrow C$ are two morphisms. There is a canonical map $u:D\to C$ from $D$ ...
7
votes
1answer
1k 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 ...
4
votes
1answer
96 views

How to get a geometric morphism out of a section? (And general pedagogy on classifying toposes)

Let $\mathcal{E}$ and $\mathcal{F}$ be toposes, $X$ an object of $\mathcal{E}$ and $p: \mathcal{E}/X \rightarrow \mathcal{E}$ the canonical geometric morphism (whose inverse image part is pullback ...
5
votes
2answers
228 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 ...
6
votes
0answers
142 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, ...
3
votes
0answers
173 views

Does the restriction functor (big to small) commute with the inverse functor?

I would like to know whether the restriction functor commutes with the inverse image functor. (I basically follow the terminology in "Topological and Smooth Stacks".) Let $X$ be a topological space. ...
6
votes
1answer
236 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 ...
5
votes
3answers
337 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, ...
6
votes
1answer
128 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 ...
4
votes
0answers
99 views

Is $(f_*A)\times B\to f_*(A\times f^*B)$ an iso in a elementary topos?

One can easily show by adjunction-nonsense that if we are given toposes $\cal E,F$ and a geometric morphism $f\colon \cal E\to F$ then there exist canonical arrows $$ \begin{gather} (f_*A)\times ...
19
votes
1answer
454 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 ...
9
votes
2answers
381 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 ...
3
votes
2answers
161 views

Are abelian groups in a [elementary] topos $\mathcal E$ an abelian subcat of $\mathcal E$?

The title tells everything: an abelian group object in a category $\mathbf C$ with finite products is a triple $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the well known diagrams ...
6
votes
1answer
460 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 ...
3
votes
1answer
101 views

Locating a copy of a thesis

Does anyone have a transmittable copy of the thesis "Logical and cohomological aspects of the space of points of a topos" by Carsten Butz? The link provided in ...
10
votes
1answer
655 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?
6
votes
1answer
324 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 ...
5
votes
0answers
209 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 ...
5
votes
2answers
233 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 ...
4
votes
1answer
122 views

Do functions defined on global elements give rise to arrows in a well-pointed topos?

Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements ...