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)]).
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 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.
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.
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:
Remark 6. (a) If C is atomic then kC is a unifunctor.
Remark 8. (a) Any unifunctor is uniform.
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.
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.
Remark 10. In an atomic category C the unifunctor kC plays the traditional role of underlying functor: