If T and S are two sets of maps to X we say that T is dominated by S if T S. By (2.1.1.d) T is dominated by S iff any noninitial map to X which factors through a map in T is not disjoint with S; if T is a sieve then T is dominated by S iff any noninitial map in T is not disjoint with S. Definition 2.3.1. A divisor
is a class D of maps satisfies the following conditions:
Remark 2.3.2. (a) (2.3.1.c)
is equivalent to that f*(sie(u)) is dominated by its
subset of maps in D.
Definition 2.3.3. (a) A submonic
divisor is a divisor consisting of monos.
Example 2.3.3.1. (a) The class O
of all the maps is the largest divisor, called the dense
divisor of A.
Suppose D is a divisor; a map in D is called a Dmap; a subobject determined by a Dmono is called a Dsubobject; a unipotent cover on an object X consisting of Dmaps is called a Dcover. Proposition 2.3.4. (a) Any isomorphism
is a Dcover; the empty set is a Dcover only
on the initial object 0.
Proof. (a) is obvious.
Definition 2.3.5. A (basis for a ) Grothendieck
topology on A is called unipotent
if it satisfies the following conditions:
Remark 2.3.6. (2.3.5.b) implies that any sheaf of sets on A for a unipotent Grothendieck topology sends 0 to a onepoint set. By Yoneda lemma this in turn implies that the initial object 0 in A is also the initial object of the category of sheaves on A. Proposition 2.3.7. (a) A Grothendieck
topology is unipotent iff any nonisomorphic initial map is not a cover
and the empty set is a cover only on 0.
Proof. (a) The condition is clearly necessary by (2.2.3.e).
Conversely, assume the condition holds for a Grothendieck topology. Suppose
{u_{i}: U_{i} > X} is a cover for the topology
on an object X. If f: Y > X is any map, by the condition
of a Grothendieck topology (see (2.3.4.b)),
there is a cover {v_{j}: V_{j} > Y} such
that for each j, fv_{j} factors through some u_{i}.
If f is disjoint with every {u_{i}}, then each v_{j}
is an initial map, so Y is an initial object by assumption, which
implies that {u_{i}} is a unipotent cover.
Proposition 2.3.8. If D is a divisor then the collection of Dcovers form a Grothendieck topology on A. Proof. The collection of Dcovers is a Grothendieck topology by (2.3.4). It is unipotent by (2.3.7) and (2.3.4.a). Corollary 2.3.9. The collection of unipotent covers form a unipotent Grothendieck topology on A. Proof. Apply (2.3.8) to the divisor O of all the maps. Definition 2.3.10. The Grothendieck topology of unipotent covers
is called the dense topology.
