0
votes
1answer
18 views

Surjection Vs Surjective geometric morphism

Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is ...
2
votes
1answer
49 views

Every Presheaf Is Colimit of Representables

I am working on the proof that each presheaf $F\in \mathbf{Set}^{\mathcal C^{op}}$ is the colimit of $\mathbf{y}\circ \pi\colon \int_{\mathcal{C}}F\to \mathbf{Set}^{\mathcal C^{op}}$ where ...
2
votes
0answers
45 views

Why $F^{+}$ is a monopresheaf?

I'm having difficult in proving that given a site $(C, J)$, then $F^{+}(c) = colim_{R \in J(c)} R $ is a monopresheaf, in the sense that there exists at most one lifting of $R \longrightarrow F^{+}$ ...
9
votes
1answer
100 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 ...
2
votes
1answer
59 views

Is the subobject classifier of the sheaves the sheaffication of the one from the presheaves?

maybe this is an idiot question, but I could not solve it. Let $\Omega$ be the suboject classifier in the category $\mathbf{PSh}(X, J)$ where $(X, J)$ is a site, I know that $\Omega(U) \cong Nat(h_U, ...
2
votes
2answers
57 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 ...
2
votes
1answer
73 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 ...
2
votes
1answer
74 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
72 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
140 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 ...
1
vote
2answers
65 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
79 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
155 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
85 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
70 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
160 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
106 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 ...
3
votes
2answers
65 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 $\mathcal G$ ...
1
vote
0answers
54 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 ...
4
votes
1answer
86 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
379 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 ...
2
votes
1answer
49 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
54 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 ...
1
vote
1answer
81 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} ...
0
votes
1answer
204 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
112 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
100 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 $f:X\rightarrow Y$ ...
3
votes
0answers
155 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. ...
4
votes
0answers
94 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 ...