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Atomic Categories
Zhaohua Luo
(Revised 9/29/98)
A category is called left if it has a strict initial object. Consider a left category C with a strict initial 0. A map is called non-initial if its domain is not initial. A class of objects (or a full subcategory) B of C is called unidense if for any non-initial object X in C there is a map from a non-initial object in B to X. By a full left dense subcategory of C we mean a full dense subcategory of C containing 0. Note that any left dense full subcategory of C is unidense. 

If D is another category  with a strict initial object, a functor F: D --> C is called nondegenerate if for any object X in D, F(X) is initial iff X is initial. 

Definition 1. (a) A non-initial object T is called unisimple if for any two non-initial maps f: X --> T and g: Y --> T there are two non-initial maps r: R --> X and s: R --> Y such that fr = gs (cf. [Luo 1998, (3.3.5)]). 
(b) is called atomic if the class of unisimple objects is unidense. 
(c) C is called unisimple if any non-initial object is unisimple. 

Denote by S(C) the full subcategory of unisimple objects of C. Adding the initial object 0 to S(C) we obtain a category S*(C) with 0 as a strict initial object. Note that in general  S*(C) is not unisimple. 

Proposition 2. (a) Any unisimple category is atomic. 
(b) If C is atomic then S*(C) is unisimple. 
(c) C is atomic if it has a full unidense atomic subcategory. 
(d) C is atomic if it has a full left dense atomic subcategory. 

Proof. (a) and (d) are obvious; (b) can be verified directly. 
(c) If B is a full unidense atomic subcategory of C then S(B) Í S(C). Since B is atomic, S(B) is unidense in B. By assumption B is unidense in C, thus S(B) is unidense in C. It follows that S(C) is also unidense in C
(e) follows from (c) and (d). 

Denote by SET the metacategory of sets. Let S*: S*(C) --> SET be the functor sending 0 to the empty set and each non-initial object in S*(C) to a one point set. Let kC: C --> SET be the Kan extension of S* to C. The functor kC is uniquely determined by C up to equivalence. If kC(X) is small for each object X in C then kC is regarded as a functor from C to the category Set of small sets. 

Example 2.1. For any object X one can define kC(X) directly: an element of kC(X) is represented by a map p: P --> X from a unisimple object P to X. If q: Q --> X is another such map then p and q represent the same element of kC(X) iff there are two maps r: R --> P and s: R --> Q such that pr = qs

Proposition 3. C is atomic iff kC is nondegenerate. 

Proof. By (2.1) kC(X) is non-empty iff there is a map from a unisimple object to X. This implies that S(C) is unidense iff kC is nondegenerate. 

Theorem 4. If B is a full unidense atomic subcategory of C then C is atomic and kC is the Kan extension of kD

Proof. The first assertion has been noticed in (2.c). For the second assertion note that by definition kC is the Kan extension of kS*(C). Clearly S*(B) is a full unidense unisimple subcategory of S*(C), thus kS*(C) is trivially the Kan extension of kS*(B). It follows that kC is the Kan extension of kS*(B). By definition kB is the Kan extension of kS*(B). This implies that kC is the Kan extension of kB

Definition 5. A functor T from C to the category of sets is called a unifunctor if the following conditions are satisfied: 
(a) T(X) is empty iff X is an initial object. 
(b) For any element p of T(X) there is a map t: P --> X in A such that T(P) has only one element and T(t)(T(P)) = p
(c) For any two non-initial maps f: P --> X and g: Q --> X in A such that T(f)(T(P)) = T(g)(T(Q)) there are non-initial maps u: O --> P and v: O --> Q such that fu = gv

Remark 6. (a) If C is atomic then kC is a unifunctor. 
(b) One can show easily that a category C is atomic iff it carries a functor unifunctor T to the category of sets and T is equivalent to its unifunctor kC

Definition 7. A functor F: D --> C between atomic categories is called uniform if kCF is equivalent to kD

Remark 8. (a) Any unifunctor is uniform. 
(b) A uniform functor between atomic categories is nondegenerate. 
(c) A composite of uniform functors between atomic categories is uniform. 
(d) If F: D ® C is a uniform functor then a map f in D is unipotent iff F(f) is unipotent in C

Definition 9. An atomic category with a faithful unifunctor is called a uniconcrete category. 

In practice almost all the natural metric sites arising in geometry have atomic categories as the underlying categories and the unifunctors as the underlying set-theoretic functors for the metric topologies (but the topologies may vary). Thus in these cases the "underlying structures" are intrinsic. This is perhaps a starting point of categorical geometry. Here are some examples: 

Example 9.1. Suppose C has a terminal object 1 and 1 is unidense in C. Then 1 is unisimple and C is atomic with homC(1, ~) as the unifunctor; furthermore if C is reduced then it is also uniconcrete (cf. Reduced Category)   This covers many natural atomic categories, such as the left categories of sets, topological spaces, posets, coherent (i.e. spectral) spaces, Stone spaces. In fact, all of these categories are reduced, therefore uniconcrete. 

Example 9.2. (a) The opposite of the category FAlg/k of finitely generated algebras over a field k is atomic. 
(b) The category of affine algebraic varieties over a field k is isomorphic to (FAlg/k)op, thus is atomic. 
(c) The category of reduced affine algebraic varieties over an arbitrary closed field is a reduced category, therefore also uniconcrete. 

Example 9.3. The category of locales is not atomic. 

Example 9.4. Denote by CRing the category of commutative rings (with unit and unit preserving homomorphisms). A ring is unisimple in CRingop iff it has exactly one prime ideal (thus any field is unisimple). It is easy to see that the class of fields is unidense in CRingop. Thus CRingop is atomic (but not uniconcrete). Since the category ASch of affine schemes is equivalent to CRingop, ASch is also atomic. 

Example 9.5. (a) Since ASch is a full unidense (in fact, a dense) subcategory of the category Sch of schemes, it follows from (2.c) and (9.4) that Sch is atomic. 
(b) A ringed space is unisimple iff its underlying space is a one point set. Since the class of such ringed spaces is unidense, the category RSpa of ringed spaces is atomic. Similarly the category LSpa of local ringed spaces is atomic. 

Remark 10. In an atomic category C the unifunctor kC plays the traditional role of underlying functor:  
(a) A map f is unipotent iff kC(f) is surjective (by (Remark 8.d)). 
(b) A mono f: U --> X is normal iff kC(f): f(U) --> f(X) is an embedding (i.e. an effective mono for kC, where kC is viewed as a functor to the category of discrete spaces). 
For instance in an atomic analytic category (or any atomic category with a strict initial object and finite limits, see Reduced Categories) one can define the notion of a reduced object in terms of kC: an object is reduced if any surjective map f (which means kC(f) is surjective) is epi.