4. Complete Framed Sites 

If B is a full subsite of a framed site C we denote by C(B) the set of objects X of C such that the collection of open effective sieves U on X with U Î B is an open cover of G(X). We say B is a base of C if C(B) = C

Remark 4.1. Suppose B is a full subsite of C. Then 
(a) C(B) is also a full subsite of C
(b) If C is strict then both B and C(B) are strict by (3.2). 
(c) If C(B) is strict then B is strict which is a full dense subcategory of C(B). 

A glueing diagram ({Xi}, {Uij}, {uij}) of an effective framed site C consists of a small set {Xi} of objects of C together with, for any i ¹ j, an open subobject Uij of Xi and an isomorphism of open subobjects uij: Uij ® Uji, such that 
(i) uji = uij-1
(ii) uij(Uij Ç Uik) = Uji Ç Ujk
(iii) uik = ujkuij on Uij Ç Uik
A glueing colimit of a glueing diagram ({Xi}, {Uij}, {uij}) is an object X of C, together with open effective morphisms vi: Xi ® X for each i, such that {Xi} covers X, with Uij = Xi Ç Xj as subobjects of X, and vi = vjuij on Uij (if Uij are all empty then we say that X is the disjoint joint of the Xi). Note that since {Xi} covers X and C is strict, X is a colimit of the glueing diagram  ({Xi}, {Uij}, {uij}), therefore is uniquely determined up to isomorphism. 
Definition 4.2. (a) A complete framed site is a strict effective framed site C such that any glueing diagram of C has a glueing colimit. 
(b) A completion of a strict framed site B is a complete framed site containing B as a base. 

Remark 4.3. If C is a locally small complete framed site and B a full subsite of C then C(B) is a completion of B

Now suppose B is a locally small strict framed site. Denote by B^ and B~ the categories of presheaves and sheaves of sets on B respectively. Then B is a full dense subcategory of B^ and B~. Applying (3.3.a) we obtain the extension of G on B^, denoted also by G. Since B is locally small, (B^, G) is active, and is exact by (3.3.b), hence (B^, G) is an effective framed site. On the other hand it is easy to see that B~ is a strict effective framed site containing B as a subsite by (3.3.d). Note that if B is small then the framed site B~ is locally small and complete. 

Theorem 4.4. Any locally small strict framed site B has a completion, which is unique up to equivalence. If B has products or fibre products then so does any completion of B

Proof. Denote by C ' the full subcategory of C~ consisting of the sheaves X on C with a small frame G(X). Then C ' is a locally small complete framed site containing C as a dense strict exact subsite. Applying (4.3) we see that the completion C(C) of C in C ' is a completion of C. The other assertions follow exactly as in the case of metric sites (see [Luo 1995a, (2.5), (2.9) and  (3.10)]). n 

Definition 4.5. Any sheaf in the completion C(C) of C ' is called a Dedekind cut of the site C

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