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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 ...
4
votes
1answer
154 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
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 ...
2
votes
1answer
51 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
56 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
83 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
330 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
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 ...
0
votes
1answer
220 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 ├ętale or fppf. Let $G \rightarrow H$ be a morphism of abelian sheaves on this ...
0
votes
1answer
119 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
107 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
136 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
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 ...
4
votes
1answer
95 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
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 ...
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, ...
3
votes
0answers
161 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
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 ...
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, ...
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 ...
4
votes
0answers
98 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
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 ...
9
votes
2answers
351 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
154 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 ...
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 ...
3
votes
1answer
99 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
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?
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 ...
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 ...
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 ...
4
votes
1answer
117 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 ...