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:
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. quasicompact, 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:
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.
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.
1.1.18 Let S be a fixed object of a category C. An Sobject (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 Sobject, a morphism h: (X, f) ® (Y, g) of Sobjects is a morphism h: X ® Y such that gh = f, and is often called an Smorphism. Denote by C_{/S} the category of Sobjects. We have a natural projection functor p: C_{/S} ® C sending each Sobject (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 kvariety 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 kalgebra. A kprevariety is a reduced local ringed space (X, O) over k covered by open sets U such that (U, O_{U}) is an affine kvariety. The category PVar/k of kprevarieties is a subpresite of the presite GSp/k of geometric spaces over k. The category AVar/k of affine kvarieties 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.
Definition 1.1.23. Suppose D is a category. By a Dset we mean a set X of objects of D. Suppose X, Y are two Dsets. A morphism of Dsets 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 Dsets. Clearly Spot(D/Set) is naturally equivalent to D. Example 1.1.24. (a) Ring^{op}/Set,
RPot/Set, and RSet 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 nonrepresentable 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 Dset {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 Dsets. Thus we obtain a covariant functor f_{D}:
C ® D/Set which is
an isogeny of presites.
