Uniform Functors    Zhaohua Luo   (7/16/1998)     In a previous note [atomic categories] we introduced the notion of an atomic category, and showed that each atomic category C carries a canonical functor kC to the category of sets, called the unifunctor of C. We also introduced the notion of a uniform functor between atomic categories. By definition a functor F: D --> C between atomic categories is called uniform if kCF is equivalent to kD. For instance, the unifunctor kC is an example of uniform functors.  A category if called left if it has a strict initial object. In this note we give an intrinsic definition of a uniform functor between any two left categories. A functor F between left categories is uniform iff it induces an isomorphism between the complete boolean algebra of normal sieves on an object and that of its image. For a set (as an object in the category Set of sets) this complete boolean algebra of normal sieves is simply its power set. This will imply that any uniform functor to Set is unique up to equivalence.  Consider a left category C with a strict initial 0. A map is called non-initial if its domain is not initial. Suppose f: Y --> X, g: Z --> X are two maps. We say that f and g are disjoint if the initial object is the pullback of f and g. We say that g dominates f if any non-initial map to X which factors through f is not disjoint with g.   Remark 1. (a) An object is initial iff its identity map with itself is disjoint.  (b) If f factors through g then g dominates f.  (c) If g dominates f and f is non-initial then g is non-initial.  Remark 2. Suppose f: Y --> X, g: Z --> X, h: W  --> X are three maps.  (a) Suppose g dominates f and f dominates h, then g dominates h.  (b) If g dominates f and h is disjoint with g, then h is disjoint with f.  Example 2.1. (a) In the category of sets (resp. topological spaces), a map g: Z --> X dominates a map f: Y --> X iff g(Z) contains f(Y).  (b) More generally, if C is atomic then a map g: Z --> X dominates a map f: Y --> X iff kC(g)(kC(X))  contains kC(f)(kC(Y)).  Definition 3. A functor F: D --> C from a left category D to a left category C called (left) uniform if the following conditions are satisfied:  (a) Two maps f and g in D with the same codomain are disjoint iff F(f) and F(g) are disjoint.  (b) Any non-initial map t: T --> F(X) with X D dominates a non-initial map F(s) with s: S --> X in D.  Proposition 4. Suppose F: D --> C is a uniform functor. Suppose f: Y --> X, g: Z --> X are two maps. Then g dominates f iff F(g) dominates F(f).  Proof. We may assume f is non-initial.   First assume F(g) dominates F(f). Consider a non-initial map h: W --> X which factors through f. Then the non-intial map F(h) factors through F(f). Thus F(h) is not disjoint with F(g), which implies that h is not disjoint with g. This implies that g dominates f.  Conversely, assume g dominates f. Consider a non-initial map h: W --> F(X) which factors through F(f). Then h dominates a non-initial map F(s) with a non-initial map  s: S --> X in D. We have to prove that F(g) is disjoint with h. Assume this is not the case, i.e. F(g) is disjoint with h. Then by (2.b) F(g) is disjoint with F(s). Since F is uniform, this implies that g is disjoint with s. Since g dominates f by assumption, f is disjoint with s. On the other hand F(f) dominates h and h dominates F(s) implies that F(f) dominates F(s). By (a) we see that f dominates s, which is a contradiction as s is non-initial. This proves that F(g) dominates F(f).  Definition 5. A functor F: D --> C between left categories is called (left) nondegenerate provided that F(X) is initial iff X is initial for any object X in D.  Proposition 6. (a) Any equivalence between left categories is uniform.  (b) Any uniform functor is nondegenerate.  (c) A composite of uniform functors is uniform.  Proof. (a) is obvious.  (b) Suppose F: D --> C is a uniform functor. If X is an initial object and 1X is its identity map then 1X is disjoint with itself iff 1F(X) is disjoint with itself. It follows from (1.a) that X is initial iff F(X) is initial.  (c) Suppose F: D --> C and G: E --> D are two uniform functors between left categories. Suppose t: T --> FG(X) is a non-initial map with X E. Since F is uniform, t dominates a non-initial map F(s) with s: S --> G(X) in D. Since G is uniform, s dominates a non-initial map G(r) with r: R --> X in E. By Prop. 4 the map F(s) dominates FG(r). It follows that t dominates FG(r) by (2.a). This proves (3.b) for FG. Clearly (3.a) is satisfied by FG. Thus FG  is uniform.  Example 6.1. (a) If B is a uni-dense full subcategory of C then the inclusion B --> C is uniform.  (b) If B is a full dense subcategory of C containing 0 then it is uni-dense, therefore by (a) the inclusion B --> C is uniform.  Example 6.2. A functor F: C --> Set is uniform iff the following conditions are satisfied:  (a) F is nondegenerate.  (b) For any element x in F(X), where X is an object in C, there is a map f: Y --> X in C such that F(f)(F(Y)) = x. (c) Use the notation of (b), if g: Z --> X  is another map in C such that F(g)(F(Z)) = x, then f and g are not disjoint.  Example 6.3. The uni-functor on an atomic category is uniform.  Example 6.4. The Zariski topology on the category of affine schemes (resp. schemes) is a uniform functor (to the category of topological spaces). In fact, most of the natural metirc sites arising in geometry have uniform metric topologies.  Example 6.5. Every frame is isomorphic to a subframe of a complete boolean algebra (see [Johnstone 1982  p.53, Cor 2.6]). The category CBoolop of boolean locales is a uni-dense full subcategory of the category Loc of locales. Thus the inclusion  CBoolop --> Loc is uniform.  If S is a set of maps to an object X we denote by S the sieve of maps to X which is disjoint with each map in S. The set S is called a unipotent cover on X if S consists of only initial map. We say S is a normal sieve if S =  S. A map is called unipotent if it is a unipotent cover. A mono is called normal if it generates a normal sieve. If C has pullbacks then a mono is normal iff any of its pullback is not proper unipotent. The class of unipotent (resp. normal) maps is closed under compositions and stable, and any intersection of normal monos is normal. Geometrically a unipotent map (resp. normal mono) plays the role of a surjective map (resp. embedding).  Denote by C(X) (or simply (X)) the set of normal sieves on an object X. (X) is a complete boolean algebra with = . Consider a map f: Y --> X. If S is a set of maps to X we denote by f*(S) the inverse image of S under f, which consists of all the maps z: Z --> Y such that f°z is in S. If S is a sieve on X then f*(S) is a sieve on Y, and we have  f*( S) = f*(S) for any sieve S on X. If S is normal then f*(S) is normal. Thus obtain a function  f*: (X) --> (Y) preserving intersections, which is a morphism of complete boolean algebras. It follows that is a functor from C to the metacategory BLOC of boolean locales, called the boolean functor on C.  If T and S are two sets of maps to X we say that S dominates T if T   S. It is easy to see that S dominates T 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 S dominates T iff any non-initial map in T is not disjoint with S. If T and S each consists of a single map we obtain the notion of a map dominates another maps defined earlier.  Remark 7.  (a) A sieve U on an object X is normal iff U contains any map to X dominated by U.  (b) Suppose F: D --> C is a uniform functor and T and S are two sets of maps to an object X in D. Then S dominates T iff F(S) dominates F(T) (the proof is similar to that of Prop. 4).  Suppose F: D --> C is a functor between left categories. If X is an object of D and V is a sieve on an object F(X) we write FX-1(V) for the sieve of maps s on X such that F(s) V.  Proposition 8. A functor F: D --> C between left categories is uniform iff the following conditions are satisfied for any object X of D:  (a) FX-1(V) is a normal sieve for any normal sieve V on an object F(X).  (b) FX-1: (f(X)) --> (X) is bijective (thus an isomorphsim of complete boolean algebras).  Proof. If u: U --> X is dominated by FX-1(V) then F(u) is dominated by F(FX-1(V)) V. Since V is normal, we have F(u) V, thus u FX-1(V). This shows that FX-1(V) is normal by (7.a).  Clearly FX is injective because by (3.b) F(FX-1(V)) dominates and is dominated by V. On the other hand, if U is a normal sieve on X then U dominates FX-1(  F(U)) by (7.b), thus U = FX-1(  F(U)) as U is normal. This shows that FX-1 is onto.  Coroally 9. A functor F: D --> C between left categories is uniform iff CF is equivalent to D.  Theorem 10. (a) The boolean algebra for an object in a locally atomic category is atomic.  (b) Any uniform functor from a left category category to the category Set of sets is unique up to equivalence.    Proof. First note that if Z is a set in Set then any normal sieve on Z is determined by a subset. This implies that (Z) is isomorphic to the power set of Z, i.e. (Z) is a complete atomic boolean algebra.  Consider a uniform functor F: D --> Set.  (a) By Prop. 8 for any object X, FX-1: (f(X)) --> (X) is an isomorphsim of complete boolean algebras. Since f(X) is a set in Set, (f(X)) is atiomic, so is (X).  (b) From (a) we see that F is equivalent to the composite of : D -->  CABoolop and the isomorphism CABoolop --> Set, where CABool is the category of complete atomic boolean algebras.  Definition 11. (a) A left category is called everywhere effective if every normal sieve is generated by a normal mono.  (b) A left category is locally atomic if the complete boolean algebra (X) of each object X is atomic (cf [Luo 1988, (2.4.5)).  Proposition 12. (a) An everywhere effective left category is atomic iff it is locally atomic.  (b) A left category is locally atomic iff there is a uniform functor to the metacategory of sets.  (c) Any subnormal framed topology on an atomic (or locally atomic) category is spatial.  Proof. (a) Suppose D is a locally atomic everywhere effective category. Suppose X is a non-intial object. Then the complete boolean algebra (X) is atomic. Suppose P is an atom of (X). Then it is generated by a non-initial normal mono v: V --> X. Since P is a minimal non-initial sieve, any two non-intial maps to V are not disjoint. Thus V is unisimple. This shows that D is atomic.  (b) follows from Theorem 10, and the fact that the category of atomice boolean locales is isomorphic to the category of sets..  (c) has been proved in [Luo 1988 (2.6.5)].  Example 12.1. The atomic categories of sets, topological spaces, posets, ringed spaces, local ringed spaces are all everywhere effective.  Example 12.2. Suppose C is a Grothendieck topos.  (a) For any object X denote by (X) the set of subobjects of X. Then (X) is a locale, and the subset   (X) of subobjects U of X such that U =  U is a boolean locale isomorphic to C(X) (see [Borceux 1994, Vol. III, p.11, (1.2.13)]). Thus C is everywhere effective.  (b) It follows from (a) that a functor F: D --> C between two Grothendieck toposes is uniform iff for any object X in D the functor F induces an isomorphism from   (X) to   (F(X)) (sending each subobject u: U ® X to the image of F(u)).  (c) It follows from (a), (b) and Prop. 12 the following assertions are equivalent:  (i) C is atomic;  (ii) C is locally atomic;  (iii) The boolean locale   (X) for each object X is atomic.  (iv) There is a uniform functor to the category Set of sets.        [References][Notations][Home]  