2.3. Divisors   If T and S are two sets of maps to X we say that T is dominated by S if T Í ØØS. By (2.1.1.d) T is dominated by S iff any non-initial 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 non-initial map in T is not disjoint with S.  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 {ui: Ui ® X} is a D-cover. If f: Y ® X is any map there is a D-cover {vj: Vj ® Y} such that for each j, f°vj factors through some ui.  (c) Suppose {ui: Ui ® X} is a D-cover on X. Suppose for each i one has a D-cover {uij: Uij ® Ui}. Then the collection {ui°uij: Uij ® 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 ui 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 {ui: Ui ® 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 {vj: Vj ® Y} such that for each j, f°vj factors through some ui. If f is disjoint with every {ui}, then each vj is an initial map, so Y is an initial object by assumption, which implies that {ui} 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.    [Next Section][Content][References][Notations][Home]