Definition 6.3.1. A simple site is a strict site C having the following properties: 
(a) Any point and any closed subset of an object is effective. 
(b) Any section between spots is an isomorphism. 
(c) Two morphisms f, g: X ® Y are equal if for any x Î |X| we have f(x) = g(x), and f_x = g_x, where f_x: x ® f(x) and g_x: x ® g(x) are morphisms of spots. 

Example 6.3.2. Set, Top, GSp, GSet, GASch, PVar/k and AVar/k are simple sites. 

Proposition 6.3.3. Suppose C is a simple site with finite limits. 
(a) Any section g: S ® X is effective (thus any diagonal or graph morphism is effective). 
(b) Any section g: S ® X is universally closed if and only if its image g(S) is a closed set. 
(c) If C is everywhere effective, then any regular monomorphism is effective. 

Proof. (a) Suppose g: S ® X is a section of a morphism f: X ® S. Since g is bicontinuous, to prove that g is effective, it suffices to prove that it is active. Suppose h: Z ® X is a morphism with h(Z) Í g(S). We have to prove that h factors through g. Consider the composition gfh: Z ® X ® S ® X. It suffices to prove that g(fh) = h. For any z Î |Z| write h(z) = z' and fh(z) = z". Then f_z'g_z" = 1_z", which implies that g_z" is a section. Since C is simple, g_z" must be an isomorphism (6.3.1.b), therefore we have g_z"f_z' = 1_z', hence (g(fh))_z = g_z"f_z'h_z = h_z for any z Î |Z|. It follows that g(fh) = h (6.3.1.b). This shows that g is effective. 
(b) One direction is obvious. Suppose |g(S)| is a closed set. Suppose h: Z ® X is any morphism. Then the image of |p|: |Z ×_X S| ® |Z| is the closed set |h^-1(g(S))| (5.2.3.e). It suffices to prove that p is bicontinuous. Since |h^-1(g(S))| is effective (6.3.1.a) and p is quasi-effective (1.2.5.d), it follows that p is effective (1.2.5.e), hence bicontinuous. 
(c) Any regular monomorphism can be obtained from a section by base extension (5.1.10.c). The proof of (b) shows that any special morphism p is effective. 

Corollary 6.3.4. Suppose C is an everywhere effective simple site with finite limits. Then any special morphism s: Z ® W is uniquely determined by the image |s(Z)| of |Z| in |W|. In particular, we have 
(a) Suppose f: X ® Y is any morphism. Then the diagonal morphism D_f: X ® X ×_S X is uniquely determined by the subset |D_f(X)| of |X ×_S X|. 
(b) Suppose i, j: Y ® X are two S-morphisms with the kernel p: ker (i, j) ® Y. Then p is uniquely determined by the subset |p(ker (i, j))| of |Y|. 
(c) Any S-section s: S ® X of an S-object f: X ® S is uniquely determined by the subset |s(S)| of |S|. 
(c') Suppose f: Y ® X is an S-morphism. Then the graph morphism G_f = (1, f): Y ® Y ×_S X is uniquely determined by the subset |G_f(Y)| of |Y ×_S X|. 

A category D is called simple if it is a simple site as a site of spots. Thus D is simple if and only if any section in D is an isomorphism. (If D has fibre products then this is equivalent to the assertion that any regular monomorphism is an isomorphism.) If D is a simple category, the site D/Set of D-sets is an everywhere effective simple site. Applying (6.3.3.c) we see that any regular monomorphism in D/Set is effective. 

Example 6.3.5. If C is a simple site, then Spot(C) is a simple category. 

Example 6.3.6. The most important simple category is the category GPot = Field^op of geometric points.  

Suppose C is a strict metric site. By a simple model of C we mean a full, simple, isometric coreflective subsite C' of C.  

Proposition 6.3.7. Suppose C is a strict metric site with finite limits. Suppose C' is a simple model of C. Then 
(a) C' has finite limits and the coreflector J: C ® C' preserves finite limits. 
(b) Any section g: S ® X in C is universally closed if and only if its image |g(S)| is a closed set. 
(c) If C' is everywhere effective, then any regular monomorphism in C is universally bicontinuous. 
(d) A morphism f: X ® Y in C is (universally) closed, separated, quasi-separated, quasi-compact if and only if the associated morphism J(f): J(X) ® J(Y) is so in C'. 

Proof. (a) follows from (4.2.9). (b) and (c) hold for the simple site C', therefore also hold for C because J: C ® C' is an isometry preserving finite limits by (a). By the same reason (d) holds.  

Example 6.3.8. GSp is a simple model of LSp, GSet is a simple model of LSet, and GPot is a simple model of LPot. 

We now study the universally injective morphisms in a metric site C with finite limits. Recall that any monomorphism in a metric site is universally injective. In a simple site we shall show that the converse is also true. First we make some observations. 

Suppose f: X ® Y is a universally injective morphism. Then the projection q: X ×_Y X ® X as a result of base extension is injective. Since the diagonal morphism D_f: X ® X ×_Y X is always injective (bicontinuous) and qD_f = 1_X, we see that D_f and q both are bijective. Thus we have: 

Proposition 6.3.9. Suppose f: X ® Y is a universally injective morphism. Then the projection q: X ×_Y X ® X and the diagonal morphism D_f: X ® X ×_Y X are both bijective.  

Suppose f: X ® Y is a morphism. For any object T, there is a map f_T: X(T) ® Y(T) of points with value in T (see (1.1.21)) determined by f. Then f is a monomorphism if and only if f_T is injective for any T Î C.  

Suppose C is a simple site. Then Spot(C) is a simple generic subpresite of C. Any object X determines a Spot(C)-set f(X) (1.1.28)) consisting of spots of X.  

Proposition 6.3.10. Suppose C is a simple site. Suppose f: X ® Y is a morphism. Then the following conditions are equivalent: 
(i) f: X ® Y is a monomorphism. 
(ii) f(f): f(X) ® f(Y) is a monomorphism in Spot(C)/Set (1.1.23) where f(f) is the morphism determined by f (1.1.28)). 
(iii) D_f: X ® X ×_Y X is an isomorphism.  
(iv) D_f(f): f(X) ® f(X) ×_f(Y) f(X) is an isomorphism.  
(v) For any spot T, f_T: X(T) ® Y(T) is injective.  
(vi) f is injective, and for any x Î X, the associated morphism x ® f(x) is a monomorphism in Spot(C). 
(vii) f is universally injective. 
(viii) D_f: X ® X ×_Y X is bijective. 

Proof. The equivalences of (i) - (vi) follow from (6.3.1). Since C is separable, any monomorphism is universally injective, thus (i) implies (vii). That (vii) implies (viii) follows from (6.3.9). We now prove that (viii) implies (iv). Note that if D_f: X ® X ×_Y X is bijective, then D_f(f): f(X) ® f(X) ×_f(Y) f(X) is bijective. Since Spot(C) is simple, the section D_f(f) is an isomorphism, hence (iv) holds.  

Remark 6.3.11. Suppose D is a generic subsite of a Cauchy-complete site C. Then D/Set is naturally an isogenous coreflective subsite of C. Thus a morphism f is universally injective if and only if the morphism f(f) is universally injective in D/Set.  

Corollary 6.3.12. Suppose C is a Cauchy- complete site and D a simple, generic subsite of C. The following conditions are equivalent for a morphism f: X ® Y in C: 
(i) f(f): f(X) ® f(Y) is a monomorphism in D/Set, where f(f) is the morphism determined by f. 
(ii) D_f(f): f(X) ® f(X) ×_f(Y) f(X) is an isomorphism. 
(iii) For any spot T Î D, f_T: X(T) ® Y(T) is injective.  
(iv) f is injective, and for any x Î |X|, the associated morphism f(x) ® f(f(x)) is a monomorphism in D. 
(v) f is universally injective. 
(vi) D_f: X ® X ×_Y X is bijective. 

Proof. Note that D/Set is a simple site and Spot(D/Set) is equivalent to D. The assertions then follow from (6.3.10) by (6.3.11). 

Example 6.3.13. A morphism spot F ® spot G in GPot = Field^o is a monomorphism if and only if the field extension F of G is a radical extension.  

We say a morphism f: X ® Y of local ringed spaces is radical if it is injective, and the induced homomorphism k_f(x) ® k_x of residue fields for any point x Î X is radical.  

Example 6.3.14. Since GSp is a geometric site and Spot(GSp) = GPot, applying (6.3.10) we see that the following are equivalent for any morphism f in GSp: 
(a) f is monomorphic. 
(b) f is universally injective. 
(c) f is radical.  

Example 6.3.15. Since GPot is a generic subsite of LSp and is simple, the following are equivalent for a morphism f in LSp by (6.3.12):  
(a) f is universally injective. 
(b) f is radical.