Grothendiek Topologies – Part II

Now, although pretopologies generalize the notion of open coverings, we cannot recover the pretopology by specifying exactly which presheaves are sheaves, for the notion of a pretopology being “finer” than another is as well behaved as in the set theoretical case. For instance, if a presheaf $\mathcal{P}$ satisfies the sheaf axiom for a covering family $K$ , it will also satisfy it for any covering family $L$ that contains $K$ , and the converse is true if every morphism of $L$ factors through one of $K$ . In order to avoid this we turn to coverings that are downward saturated, in other words, sieves.

Definition: Let $\mathcal{C}$ be a category. A Grothendieck topology on $\mathcal{C}$ is given by associating to each $U \in \mathcal{C}$ a collection of $J(U)$ of $U$ -sieves such that

(i) The maximal sieve $M_U$ is in $J(U)$ .

(ii) If $S \in J(U)$ and $f: V \to U$ is a morphism of $\mathcal{C}$ , then $f^*S$ is in $J(V)$ .

(iii) If $S \in J(U)$ and $Q$ is a $U$ -sieve such that $f^*Q \in J(V)$ for any $f:V \to U$ in $S$ , then $Q \in J(U)$ .

A category with a Grothendieck topology $(\mathcal{C},J)$ is called a site.

Just as in point set topology, we can define a basis to generate a topology on a given space, a Grothendieck pretopology induces a Grothendieck topology, and it is usualy easier not only to specify the former rather than the sieves it generates, but also to verify the sheaf axiom via pretopologies when the category in questions has pullbacks. I will formally state these facts, although I haven’t really gotten into the details of sheaves on sites (but will do shortly).

Proposition 1: Let $\mathcal{C}$ be a category with pullbacks. Every Grothendieck pretopology $K$ on $\mathcal{C}$ induces a Grothendieck topology $J$ on $\mathcal{C}$ by setting

$S \in J(U) \Leftrightarrow \exists R \in K(U); R \subset S.$

In this case, we say that $K$ is a basis for $J$ .

Proposition 2: Let $\mathcal{C}$ be a category with pullbacks and $J$ a Grothendieck topology on $\mathcal{C}$ with basis $K$ . A presheaf $\mathcal{P} : \mathcal{C}^{op} \to \textbf{Set}$ is a $J$ -sheaf if, and only if, it is a $K$ -sheaf.

Proofs can be found in SGL, although the definition of basis is slightly different, and in my dissertation…

Given two Grothendieck topologies $J, J'$ on a category $\mathcal{C}$ we say that $J$ is finer than $J'$ if $J'(U) \subseteq J(U)$ for every $U \in \mathcal{C}$ .

Thus the Grothendieck topologies on $\mathcal{C}$ are partially ordered. The topology in which every sieve is a covering is called the maximal or discrete topology, and the one where only the maximal sieves are coverings is called the minimal or trivial topology, and on this topology every presheaf is a sheaf.

Given a family $\{J_\alpha\}$ of Grothendieck topologies on $\mathcal{C}$ it is clear that $\bigcap J_\alpha$ is also a Grothendieck topology on $\mathcal{C}$ , where $\left( \bigcap J_\alpha \right)(U) = \bigcap J_\alpha(U)$ , and so $\bigcap J_\alpha$ is the greatest lower bound for $\{J_\alpha\}$ . To obtain the lowest upper bound just take the intersection of all the upper bounds for $\{J_\alpha\}$ .

In a locally small category, there exists the largest topology for which all the representable functors are sheaves, called the canonical topology. A topology is said to be subcanonical if it is smaller (coarser, or less fine) than the canonical topology.

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~ by Guilherme Frederico Lima on November 23, 2008.

Posted in Sheaves

One Response to “Grothendiek Topologies – Part II”

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Guilherme Frederico Lima said this on November 23, 2008 at 1:31 pm

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