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.
Remark 2. Suppose f: Y ®
X, g: Z ®
X, h: W ® X are three maps.
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).
Definition 3. A functor F: D ®
C from a left category D to a left category C called
uniform if the following conditions
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.
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.
Proof. (a) is obvious.
Example 6.1. (a) If B is a uni-dense full subcategory
of C then the inclusion B ®
C is uniform.
Example 6.2. A functor F: C ®
Set is uniform iff the following conditions are satisfied:
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 ØS 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 ØS consists of only initial map. We say S is a normal sieve if S = ØØS. 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 ÂC(X) (or simply Â(X)) the set of normal sieves on an object X. Â(X) is a complete boolean algebra with Ù = Ç. 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*(ØS) = Øf*(S) for any sieve S on X. If S is normal then f*(S) is normal. Thus obtain a function f*: Â(X) ® Â(Y) preserving intersections, which is a morphism of complete boolean algebras. It follows that Â 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 Í ØØS. 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.
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) Î 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:
Proof. If u: U ®
X is dominated by FX-1(V) then
F(u) is dominated by F(FX-1(V))
Í V. Since V is normal,
we have F(u) Î V,
thus u Î FX-1(V).
This shows that FX-1(V) is normal by
Coroally 9. A functor F: D ® C between left categories is uniform iff ÂCF is equivalent to ÂD.
Theorem 10. (a) The boolean algebra for an object in a locally
atomic category is atomic.
Definition 11. (a) A left category is called everywhere
effective if every normal sieve is generated by a normal mono.
Proposition 12. (a) An everywhere effective left category is
atomic iff it is locally atomic.
Proof. (a) Suppose D is a locally atomic everywhere effective
category. Suppose X is a non-intial object. Then the complete boolean
algebra Â(X) is atomic. Suppose
P is an atom of Â(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.
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.