Definition 2.3.1. A divisor is a class D of maps satisfies the following conditions: 
(a) Isomorphisms and initial maps are in D. 
(b) D is closed under composition. 
(c) If f: Y ® X is a map and u: U ® X is a map in D, any map to Y whose composite with f factors through u is dominated by the set of all the maps to Y in D satisfying the same condition. 

Remark 2.3.2. (a) (2.3.1.c) is equivalent to that f*(J(u)) is dominated by its subset of maps in D. 
(b) We say a class D of maps is stable if any pullback of a map in D exists and is again in D. If D is stable then for any f and u as in (2.3.1.c) the sieve f*(J(u)) is generated by the pullback of u along f, which is a map in D. Thus (2.3.1.c) holds for any stable D. 

Definition 2.3.3. (a) A submonic divisor is a divisor consisting of monos. 
(b) A subnormal divisor is a divisor consisting of normal monos. 
(c) A divisor E is called a subdivisor of a divisor D if E is a subclass of D. 

Example 2.3.3.1. (a) The class O of all the maps is the largest divisor, called the dense divisor of A. 
(b) If A has pullbacks then the class of M of monos is the largest (stable) submonic divisor, called the monic divisor of A. 
(c) If A has pullbacks then the class of N of normal monos is the largest (stable) subnormal divisor by (2.2.6), called the normal divisor of A. 
(d) If A is an extensive category an injection of a sum is simply called a direct mono. The class E of direct monos is a stable subnormal divisor, called the extensive divisor of A. 
(e) If A is an analytic category then the class A of analytic monos is a stable subnormal divisor by (2.2.6.f), (1.6.2) and (1.6.3), called the analytic divisor of A. 

Suppose D is a divisor; a map in D is called a D-map; a subobject determined by a D-mono is called a D-subobject; a unipotent cover on an object X consisting of D-maps is called a D-cover. 

Proposition 2.3.4. (a) Any isomorphism is a D-cover; the empty set is a D-cover only on the initial object 0. 
(b) Suppose {u_i: U_i ® X} is a D-cover. If f: Y ® X is any map there is a D-cover {v_j: V_j ® Y} such that for each j, f_°v_j factors through some u_i. 
(c) Suppose {u_i: U_i ® X} is a D-cover on X. Suppose for each i one has a D-cover {u_ij: U_ij ® U_i}. Then the collection {u_i°u_ij: U_ij ® X} is a D-cover on X. 

Proof. (a) is obvious. 
(b) The collection of all the D-maps t: T ® Y such that f_°t factors through some u_i is a D-cover by (2.3.1.c). 
The proof of (c) is similar to that of (2.2.3.c). n 

Definition 2.3.5. A (basis for a ) Grothendieck topology on A is called unipotent if it satisfies the following conditions: 
(a) Any cover is unipotent. 
(b) The empty sieve is a cover on 0. n 

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 one-point 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. n 

Proposition 2.3.7. (a) A Grothendieck topology is unipotent iff any non-isomorphic initial map is not a cover and the empty set is a cover only on 0. 
(b) Any subcanonical Grothendieck topology is unipotent. 

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, f_°v_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. 
(b) Clearly the conditions of (a) hold for any subcanonical Grothendieck topology. n 

Proposition 2.3.8. If D is a divisor then the collection of D-covers form a Grothendieck topology on A. 

Proof. The collection of D-covers is a Grothendieck topology by (2.3.4). It is unipotent by (2.3.7) and (2.3.4.a). n 

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. n 

Definition 2.3.10. The Grothendieck topology of unipotent covers is called the dense topology.