MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

While reading Demazure-Gabriel's construction of $\mathcal{S}ch$ as a full subcategory of $\mathcal{P}sh(CRing^{op})$, I've been trying to translate their exposition into the language of covering sieves and Grothendieck topologies. The requirement that a scheme be a sheaf in the Zariski topology on $CRing^{op}$ is essentially already in the language of sieves, but the requirement that there exists a cover of affines is giving me some trouble.

Question, then:

In general, is there a natural induced Grothendieck topology on $\mathcal{P}sh(\mathcal{C})$ or $\mathcal{S}h(\mathcal{C})$, where $\mathcal{C}$ is a site (not a pre-site!)?

Edit: I removed the other parts of the question regarding t-Schemes and algebraic spaces as functors of points to ask them at some other time.

share|cite|improve this question
up vote 7 down vote accepted

The answer to your first question is yes. Suppose $C$ a site.

The category $C^{\sim}$ of sheaves on $C$ simply has the canonical topology (SGA 4, Vol. 1, Exp. II, 2.5). The sheaves here are precisely the representable sheaves. (I'm ignoring questions about universes.)

The category $C^{\wedge}$ of presheaves on $C$ inherits a topology in the following manner (SGA 4, Vol. 1, Exp. II, §5). Declare a morphism $F\to G$ of presheaves on $C$ to be a covering morphism if the induced morphism $aF\to aG$ of associated sheaves is an epimorphism. Now say that a collection of morphisms $F_i\to G$ is a covering family if the morphism $\amalg F_i\to G$ is a covering morphism. This gives $C^{\wedge}$ the finest subcanonical topology such that covering families in $C$ give rise to covering families in $C^{\wedge}$. This topology on $C^{\wedge}$ can be seen as the lift of the topology on $C^{\sim}$ described above along the sheafification functor $a$.

share|cite|improve this answer
One sense in which it is induced is that there is a canonical isomorphism Sh(C) \to Sh(Sh(C)). – David Zureick-Brown Jan 8 '10 at 21:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.