Definition 1.1.1. A metric presite (or simply presite) (C, t) consists of a category C and a covariant functor t from C to the category Top of topological spaces. 

We will often write C for a presite (C, t), and call t the metric functor of C. If X is an object and f: X ® Y a morphism, we shall use the following notation: 
|X| for the underlying space t(X); 
|f| for the underlying map t(f); 
|f(X)| for the subset |f|(|X|) of |Y|; 
f(U) for |f|(U) if U Í |X|, and f^-1(V) for |f|^-1(V) if V Í |Y|. 

1.1.2 A morphism f: Y ® X in a presite C is surjective, injective, bicontinuous, homeomorphic, open, closed, etc., if the underlying map |f| of f is so. An object X of a presite C is called discrete (resp. quasi-compact, resp. connected, etc.) if the underlying space |X| of X is so. Denote by Dis(C) the full subcategory of discrete objects of C. We say C is a discrete presite if each object of C is discrete. 

Any category C together with a covariant functor from C to the category Set of sets is naturally a discrete presite, called a presite of sets. A presite (C, t) has an underlying presite of sets (C, t*), where t* is the composition of t and the forgetful functor Top ® Set. 

We say an object X of a presite C is a spot if the underlying space |X| of X is a singleton (i.e., a one point space). Denote by Spot(C) the full subcategory of spots of C. A presite C is called a presite of spots if any object of C is a spot. Any category may be viewed as a presite of spots in a natural way. 

Definition 1.1.3. Suppose (C, t) and (C', t') are two presites. Suppose J: C ® C' is a functor. We say J is an isometry (resp. isogeny) of presites if t'J (resp. t'*J) is isomorphic to t (resp. t*). If an isometry J is an isomorphism (resp. equivalence) of the underlying categories then we say that J is an isomorphism (resp. equivalence) of presites, and that the presite (C, t) is isomorphic (resp. equivalent) to the presite (C', t'). 

Example 1.1.4. Suppose (C, t) is a site, B a subcategory of C, and I: B ® C the inclusion functor. We denote by t|_B = tI: B ® Top the restriction of t on B. The presite (B, t|_B), or simply B, is called a subpresite of (C, t). The inclusion functor I is then an isometry from (B, t|_B) to (C, t). 

Example 1.1.5. Any topological space X determines a presite w(X), where the objects of w(X) are the subspaces of X with the induced topology, and the morphisms of w(X) are the inclusion maps, with the metric functor w(X) ® Top sending each U Î w(X) to itself. 

Example 1.1.6. The set W(X) of open subsets of a topological space X is a subpresite of w(X). 

Example 1.1.7. The category Top of topological spaces together with the identity functor Top ® Top is a presite. The category Set of sets is a discrete subpresite of Top. 

Example 1.1.8. For any presite C up to isomorphism there is a unique isometry (resp. isogeny) from C to Top (resp. Set), i.e., the functor t: C ® Top (resp. t*: C ® Set). 

Example 1.1.9. The category RSp of ringed spaces is a presite with the forgetful functor sending each ringed space (X, O) to the underlying space X. 

Example 1.1.10. The category LSp of local ringed spaces is a subpresite of RSp. 

Example 1.1.11. Suppose X is a local ringed space. We define the geometric radical r(X) of X to be the sheaf of ideas on X given by 

Example 1.1.12. Applying Dis and Spot to the presites RSp, LSp and GSp we obtain six discrete presites: 
RSet = Dis(RSp), called the presite of ringed sets. 
LSet = Dis(LSp), called the presite of local ringed sets. 
GSet = Dis(GSp), called the presite of geometric sets. 
RPot = Spot(RSp), called the presite of ringed points. 
LPot = Spot(LSp), called the presite of local ringed points. 
GPot = Spot(GSp), called the presite of geometric points. 

Denote by Ring, LRing and Field the categories of (commutative) rings, local rings and fields respectively. Then Ring^op, LRing^op and Field^op are equivalent to RPot, LPot and GPot respectively. 

Any ring A determines a ringed point denoted by spot A; spot A is a local ringed point (resp. geometric point) if and only if A is a local ring (resp. field). 

Example 1.1.13. Recall that a subcategory D of a category C is called reflective (resp. corefelective) if the inclusion functor I: D ® C has a left (resp. right) adjoint functor J: C ® D called reflector (resp. coreflector). If C is a presite and D a subpresite of C and J is an isometry (resp. isogeny) of presites, then we say that D is an isometric (resp. isogenous) reflective (resp. coreflective) subpresite of C. 
(a) Set is an isogenous coreflective subpresite of Top; 
(b) Top is an isometric coreflective subpresite of RSp; 
(c) RSet, LSet and GSet are isogenous coreflective subpresite of RSp, LSp and GSp respectively (see (4.1.8)). 
(d) GSp is an isometric coreflective subpresite of LSp with the coreflector red sending each local ringed space (X, O) to the geometric space red X = X_red = (X, O/r(X)) (see (4.1.7)). 
(e) LSp is a coreflective subpresite of RSp with the coreflector sending a ringed space (X, O) to the local ringed space Spec X (see (4.1.6)). 
(f) RPot is a reflective subpresite of RSp with the reflector sending each ringed space (X, O) to spot O(X). 

Example 1.1.14. (Affine Schemes) An affine scheme is a local ringed space isomorphic to a spectrum (Spec A, O) of some ring A, where Spec A is the set of prime ideals of A equipped with the spectrum topology, and O is the sheaf of rings on Spec A generated by the function Spec A_f d A_f for any f Î A. Denote by ASch the category of affine schemes. ASch is a full subpresites of LSp. 

Example 1.1.15. (Schemes) A scheme is a local ringed space (X, O) which is covered by open sets U such that (U, O|_U) is an affine scheme. Denote by Sch the category of schemes. Sch is a full subpresites of LSp. ASch is a full subpresite of Sch. 

Example 1.1.16. (Reduced Schemes) A scheme (X, O) is geometrically reduced as a local ringed space if and only if O(U) is a reduced ring for any open set U. Denote by GSch (resp. GASch) the category of reduced schemes (resp. reduced affine schemes). They are full subpresites of GSp. Note that a geometric set is a reduced scheme, hence GSet is a subpresite of GSch. Similarly GPot is a subpresite of GASch. 

Example 1.1.17. (a) GSch is an isometric coreflective subpresite of Sch. 
(b) GASch is an isometric coreflective subpresite of ASch. 
(c) GSet is an isogenous coreflective subpresite of Sch and GSch. 

1.1.18 Let S be a fixed object of a category C. An S-object (or an object over S) is a pair (X, f), where X is an object and f: X ® S is a morphism; f is called the structure morphism of X over S; for simplicity we often write X, or f, for (X, f). If (Y, g) is another S-object, a morphism h: (X, f) ® (Y, g) of S-objects is a morphism h: X ® Y such that gh = f, and is often called an S-morphism. Denote by C_/S the category of S-objects. We have a natural projection functor p: C_/S ® C sending each S-object (X, f) Î C_/S to X. If (C, t) is a presite, then (C_/S, tp) is a presite and p: C_/S ® C is an isometry of presites. 

Example 1.1.19. Let k be an algebraically closed field. An affine k-variety is a reduced local ringed spaces over k which is isomorphic to the subspace of Spec A consisting of closed points, where A is a finitely generated reduced k-algebra. A k-prevariety is a reduced local ringed space (X, O) over k covered by open sets U such that (U, O|_U) is an affine k-variety. The category PVar/k of k-prevarieties is a subpresite of the presite GSp/k of geometric spaces over k. The category AVar/k of affine k-varieties is a subpresite of PVar/k (see (3.3) for other approaches). 

1.1.20 The presites Set, Top, RSp, RSet, RPot, LSp, GSp, LSet, GSet, LPot and GPot will be called the basic presites. The presites Sch, GSch, ASch, GASch, PVar/k and AVar/k will be referred as the basic algebraic presites. 

1.1.21 An object T of a category C determines a covariant functor h'_T = hom (T, ~): C ® Set. If X is any object of C we also write X(T) for the set hom (T, X). The elements of X(T) are called the points of X with values in T. A covariant functor from a category C to Set is called representable if it is isomorphic to h'_T for some T Î C. We say a presite (C, t) is representable if the forgetful functor t*: C ® Set is representable. 

Example 1.1.22. (a) Set, Top, RSp, RSet, RPot are representable. 
(b) PVar/k and AVar/k are representable for any algebraically closed field k. 

Definition 1.1.23. Suppose D is a category. By a D-set we mean a set X of objects of D. Suppose X, Y are two D-sets. A morphism of D-sets from X to Y consists a map f: X ® Y, and for each x Î X, a morphism f_x: x ® f(x) in D. We shall denote by D/Set the category of D-sets. Clearly Spot(D/Set) is naturally equivalent to D. 

Example 1.1.24. (a) Ring^op/Set, RPot/Set, and RSet are equivalent presites; 
(b) LRing^op/Set, LPot/Set, and LSet are equivalent presites; 
(c) Field^op/Set, GPot/Set, and GSet are equivalent presites. 

1.1.25 For any presite C we define the category C_* of pointed objects whose objects are pairs (x, X), where X Î C and x Î |X|. A morphism from a pair (x, X) to (y, Y) is a morphism f: X t Y such that f(x) = y. (Note that Spot(C) is naturally a subcategory of C_*.) 

1.1.26 By a generic subpresite of a presite C we mean a full subcategory D of C consisting of spots which is a coreflective subcategory of C_*. Explicitly, this means that for any object X and any point x Î |X|, there exist a morphism f: z ® X from a spot z Î D to X with |f(z)| = x, such that any morphism f': z' ® X from a spot z' Î D to X with |f'(z')| = x factors through f uniquely. The spot z is called the spot of D determined by x, denoted f_D(x), or simply f(x), and the morphism f: z ® X is called the canonical morphism of x. 

Example 1.1.27. GPot is a generic subpresite of all the non-representable basic presites LSp, GSp, LSet, GSet, LPot and the basic algebraic presites Sch, GSch, ASch, GSch and GASch. 

1.1.28 Now suppose C is a presite and D a generic subpresite of C. An object X e C determines a D-set {f_D(x) | x Î |X|}, denoted f_D(X). (More generally if U Í |X| we let f_D(U) = {f_D(x) | x Î U}.) Any morphism f: X ® Y determines a morphism f_D(f): f_D(X) ® f_D(Y) of D-sets. Thus we obtain a covariant functor f_D: C ® D/Set which is an isogeny of presites.