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 noninitial object T is called unisimple if for any two noninitial maps f: X > T and g: Y > T there are two noninitial maps r: R > X and s: R > Y such that fr = gs (cf. [Luo 1998, (3.3.5)]). 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. Proof. (a) and (d) are obvious; (b) can be verified directly. Denote by SET the metacategory of sets. Let S*: S*(C) > SET be the functor sending 0 to the empty set and each noninitial object in S*(C) to a one point set. Let k_{C}: C > SET be the Kan extension of S* to C. The functor k_{C} is uniquely determined by C up to equivalence. If k_{C}(X) is small for each object X in C then k_{C} is regarded as a functor from C to the category Set of small sets. Example 2.1. For any object X one can define k_{C}(X) directly: an element of k_{C}(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 k_{C}(X) iff there are two maps r: R > P and s: R > Q such that pr = qs. Proposition 3. C is atomic iff k_{C} is nondegenerate. Proof. By (2.1) k_{C}(X) is nonempty iff there is a map from a unisimple object to X. This implies that S(C) is unidense iff k_{C }is nondegenerate. Theorem 4. If B is a full unidense atomic subcategory of C then C is atomic and k_{C }is the Kan extension of k_{D}. Proof. The first assertion has been noticed in (2.c). For the second assertion note that by definition k_{C }is the Kan extension of k_{S*(C)}. Clearly S*(B) is a full unidense unisimple subcategory of S*(C), thus k_{S*(C) }is trivially the Kan extension of k_{S*(B)}. It follows that k_{C }is the Kan extension of k_{S*(B)}. By definition k_{B }is the Kan extension of k_{S*(B)}. This implies that k_{C }is the Kan extension of k_{B}. Definition 5. A functor T from C to the category of sets is called a unifunctor if the following conditions are satisfied: Remark 6. (a) If C is atomic then k_{C} is a unifunctor. Definition 7. A functor F: D > C between atomic categories is called uniform if k_{C}F is equivalent to k_{D}. Remark 8. (a) Any unifunctor is uniform. 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 settheoretic 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 hom_{C}(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. 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 CRing^{op} 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 CRing^{op}. Thus CRing^{op} is atomic (but not uniconcrete). Since the category ASch of affine schemes is equivalent to CRing^{op}, 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. Remark 10. In an atomic category C the unifunctor k_{C} plays the traditional role of underlying functor:
