Suppose C is a category with a strict initial object 0. Suppose U and {Ui} are normal subobjects of X and each  Ui   U. If {Ui} is a unipotent cover of U then we say that U is a normal union of {Ui}.Note that if U is a normal union of {Ui} then U generates the normal sieve   {Ui}, thus is uniquely determined by {Ui}.

Recall that a stable divisor is a class D of maps satisfies the following conditions: 
(a) Isomorphisms and initial maps are in D. 
(b) D is closed under composition. 
(c) Any pullback of a map in D exists and is again in D.
We say D is a subnormal divisor if any map in D is a normal mono.

Suppose D is a stable divisor; a map in D is called a D-map; a subobject determined by a D-mono is called a D-subobject; a unipotent cover on an object X consisting of D-maps is called a D-cover. For any object X denote by G_D(X) the set of D-subobjects of X.

Definition 6.1.1. Suppose D is a subnormal stable divisor D: 
(a) D is effective if any set of normal monos in D has a normal union which is again in D.
(b) D is strict if an object X is a normal union of {ui: Ui    X} of D-subobjects then X is the colimit of the systems {Us   Ut   X |s, t   I}}. 
(c) D is called canonical if it is effective and strict.

Definition 6.1.2. Suppose D is a subnormal stable divisor on C.
(a) A glueing diagram ({X_i}, {U_ij}, {u_ij}) for D consists of a small set {X_i} of objects of C together with, for any i ¹ j, a D-subobject U_ij of X_i and an isomorphism of subobjects u_ij: U_ij ® U_ji, such that 
(i) u_ji = u_ij^-1; 
(ii) u_ij(U_ij Ç U_ik) = U_ji Ç U_jk; 
(iii) u_ik = u_jku_ij on U_ij Ç U_ik. 
(b) A glueing colimit of a glueing diagram ({X_i}, {U_ij}, {u_ij}) is an object X of C, together with D-maps v_i: X_i ® X for each i, such that {X_i} covers X, with U_ij = X_i Ç X_j as subobjects of X, and v_i = v_ju_ij on U_ij (if U_ij are all empty then we say that X is the disjoint joint of the X_i). Note that since {X_i} covers X and C is strict, X is a colimit of the glueing diagram  ({X_i}, {U_ij}, {u_ij}), therefore is uniquely determined up to isomorphism. 
(c) A complete divisor is a strict effective D such that any glueing diagram for D has a glueing colimit. 

Definition 6.1.3. Suppose A is an analytic category. A normal completion of A is a category C containing A as a full subcategory which satisfies the following conditions:
(a) The initial object of A is also a strict initial object of C.
We say a normal mono u: U   X in C is open if its pullback along map v: V   X with V   A exist which is a normal union of analytic subobject of V. The class of open monos is a subnormal stable divisor.
(b) The class of open monos is a complete divisor on C.
(c) Any object is a normal unoin of open subobjects determined by a mono with domain in A.

Remark 6.1.2. Suppose D is an effective divisor. 
(a) G_D(X) is a frame. 
(b) For any map f: Y   X the function G_D(X)   G_D(Y)  sending each open subobject of X to its pullback along f is a morphism of frames. 
(c) The functor G_D from C to the category of locales is an effective subnormal strict framed topology. 
(d) Conversely, the effective open maps in an effecitve subnormal strict framed topology is a canonical divisor. 
 
 
 
 
 

A canonical site is a pair (C, D) consisting of a category C with a strict initial object and a canonical topology D on C. 

Suppose (C, D) is a canonical site. A D-mono is simply called an open effective map. A D-subobject is called an open subobject. 

Definition 6.1.4. We say that a canonical site C is an extension of a strict analytic category A if the following conditions are satisfied: 
(a) A mono in A is an analytic mono in A iff it is open effective in C. 
(b) Any object X in C is the normal union of its open subobjects U determined by the open effective monos with domain in A. 
If C is a complete site then we say that C is a complete extension of A. 

Proposition 6.1.5. (a) Suppose a canonical site C is a completion of a strict analytic category A. Then C is uniquely determined by A, denoted by Sch(A), 
(b) Sch(A) has pullbacks and disjoint stable sums. 
(c) Any extension of A is equivalent to a full subcategoy of its complete extension. 
 
 

We say D is subnormal if any mono in D is normal. 

Definition 6.1.2. Suppose D is a subnormal stable divisor. 
(d) D is called effective if any set of monos to an object X in D is a D-cover on a mono to X in D. 
(e) D is called strict 
Definition 6.1.3. (a) A canonical topology is a strict, effective subnoraml stable divisor. 
 
 
 
 

Consider a fixed dense full subcategory A of C containing 0. 

Proposition 6.1.1. (a) Any normal mono in A is a normal mono in C. 
(b) A sieve on an object X is  normal in C if its pullback along any map Y ® X with Y in A is normal in C. 

analytic. 

Definition 6.1. (a) A sieve on an object X is called A-open if its pullback along any map Y ® X with Y in A is analytic. 
(b) A mono is called A-open if it generates an open sieve. 
(c) A mono is called A-affine if it is A-open with domain in A. 
(d) A subobject is called A-open (resp. A-affine) if it is determined by an A-open (resp. A-affine) mono. 

Definition 6.2. We say A is an affine geometry on C if the following conditions are satisfied: 
(a) Any object which is isomorphic to an object in A is in A. 
(b) Any object has a unipotent cover consisting of A-affine monos. 
(c) Any A-open sieve is generated by a mono. 
 
Suppose A is an affine geometry on C. 

Proposition 6.3. (a) Any initial object is in A. 
(b) Any A-open sieve is a normal sieve. 
(c) Any A-affine mono is a normal mono. 
(d) The class of A-affine monos is a stable subnormal divisor on C. 

Proof. (a) By (6.2.b) there is an A-affine mono to 0, which must be an isomorphism as 0 is strict. Thus by (6.1.a) 0 is in A. 
(b) Consider an A-open sieve U on an object X. 
 
 
 
 
 
 

initial object 0 has a unipotent cover consisting of A 
 
 
 
 

For any object X denote by G_A(X) the set of A-affine subobjects of X. 

(e) G_A(X) is a locale for each object X. 
(d) G_A defines a framed topology on C. 

is called affine if Y    A and any pullback of it along a map Z   X with Z   A generates an analytic sieve. 
 

A geometric divisor on C is a stable, subnormal, strict, effective and complete divisor. 

Let A be a strict coherent analytic category and C a category containing A as a full subcategory. 

We say C is a geometric extension of A if the following conditions are satisfied: 
(a) If U is a sieve on an object such that any its pullback along a map 
 
 
 
 
 
 
 
 

If f: Y ® X is a map and u: U ® X is a map in D, any map to Y whose composite with f factors through u is dominated by the set of all the maps to Y in D satisfying the same condition. 
 
 
 

Consider a fixed subnormal divior D. By an affine object (with respec to D)  we mean an object which is the domain of a non-isomorphic mono in D. Denote by ASch(C) the full subcategory of affine objects. 

Definition 6.1.2. An algebraic geometry is a pair (C, D) consisting of a category with a strict initial and a subnormal divisor D on C satisfying the following conditions. 
(a) ASch(C) is a coherent analytic geometry. 
(b) A mono in ASch(C) is in D iff it is an analytic mono in ASch(C). 
(c) The framed topology F_D is complete (i.e. D is complete). 
 
 

Let A be a strict coherent analytic category. We assume A is non-trivial (i.e. with a non-initial object). Recall that if X is an object the set Spec(X) of primes of X is a topological spac, called the spectrum of  X. A is a strict metric site with the metric topology Spec sending each object X to its spectrum Spec(X). Since Spec(X) is a quasi-compact space for any object X of C, the metric site A is not complete (as in a complete metric site the space of any infinite disjoint sum of non-empty objects is not quasi-compact). 

Let C be a fixed completion of the strict framed site A. Then C is a complete metric site containing A as a base. An object of C is called a scheme. If X is a scheme we denote by |X| the underlying space of X. An open subobject U of X is called affine if it is determined by an object in A.