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 $X$ ...
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
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 ...
0
votes
1answer
223 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 ...
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 ...
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, ...