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) C  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 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 non-empty 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: 
(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 k_C 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 k_C. 

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

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 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. 
(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 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. 
(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 k_C plays the traditional role of underlying functor:  
(a) A map f is unipotent iff k_C(f) is surjective (by (Remark 8.d)). 
(b) A mono f: U --> X is normal iff k_C(f): f(U) --> f(X) is an embedding (i.e. an effective mono for k_C, where k_C 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 k_C: an object is reduced if any surjective map f (which means k_C(f) is surjective) is epi.