MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 923 characters in body

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-sheaf axioms" and then use

Theorem 1: the B-sheaf extends uninquely to the whole of X.

I was wondering if there's a similar thing for more general sites, and actually not just for sheaves on a given site but for stacks.

The question I'm really interested in is the following:

If one has a fibred category over Schemes (say Schemes over some fixed field with the fppf topology) and one wants to check that descent is effective, would it be sufficient to check it on some subcategory of schemes (using perhaps some vague analogue of Theorem 1)?

Thanks.

EDIT

For example one might want to construct the stack M of coherent sheaves on some scheme X. One way to do it is to define the functor which associates with each scheme S the groupoid of coherent sheaves on $S\times X$ flat over S $$M(S) = { E \in Coh\ S\times X,\ E \text{ flat over S} }.$$

Let's say I want to use a different characterization of $M(S)$, perhaps using Lemma 3.31 on page 82 of Huybrechts' Fourier-Mukai book.

My ignorance prevents me from knowing if the that lemma is valid for a general scheme S (no matter how nice my X might be). This is why I'd like to work over some nice subcategory of schemes (where the lemma is valid) and then extend.

The stack I'd be interested in defining would be a stack of perverse sheaves on X, where the matters would be a bit worse.

1

# Does the concept of a basis for a topology on a category exist?

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-sheaf axioms" and then use

Theorem 1: the B-sheaf extends uninquely to the whole of X.

I was wondering if there's a similar thing for more general sites, and actually not just for sheaves on a given site but for stacks.

The question I'm really interested in is the following:

If one has a fibred category over Schemes (say Schemes over some fixed field with the fppf topology) and one wants to check that descent is effective, would it be sufficient to check it on some subcategory of schemes (using perhaps some vague analogue of Theorem 1)?

Thanks.