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 glueing diagram ({X_{i}},
{U_{ij}}, {u_{ij}}) of an effective framed
site C consists of a small set {X_{i}} of objects
of C together with, for any i ¹
j, an open subobject U_{ij} of X_{i}
and an isomorphism of open subobjects u_{ij}: U_{ij}
® U_{ji}, such that
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.
