6. Geometric Subsites  Suppose (C, G) is a framed site. If U is a sieve on an object X of C we denote by U* the joint of open subsieves Ui of U in G(X) (i.e. U* = Úi Ui).  Proposition 6.1. Suppose U and V are two sieves on X.  (a) Suppose V is open. Then V Í U* if and only if V has an open cover {Vi} with each Vi Í U.  (b) (U Ç V)* = U* Ç V*.  (c) If f: Y ® X is a morphism then f-1(U*) Í (f-1(U))*.  (d) If f: Y ® X is an open effective morphism then f-1(U*) = (f-1(U))*.  (e) If C is neat then U Í U*.  (  Proof. (a) If V Í U* then V = V Ç U* = V Ç (Úi Ui) = Úi V Ç Ui. This shows that {Vi = V Ç Ui | Ui is any open subsieve of U} is an open cover of V. The other direction is trivial.  (b) Clearly we have (U Ç V)* Í U* Ç V*. Since U* Ç V* Í U*, U* Ç V* has an open cover {Ui | iÎ I} with each Ui Í U by (a). Similarly U* Ç V* has an open cover {Vj | j Î J} with each Vj Í V. Then {Ui Ç Uj | i Î I and j Î J} is an open cover of U* Ç V* and each Ui Ç Vj Í U Ç V. It follows that U* Ç V* Í (U Ç V)* by (a). The other direction is trivial.  (c) Suppose {Ui} is an open cover of U* with each Ui Í U, then f-1(Ui) is an open cover of f-1(U*) and f-1(Ui) Í f-1(U). Thus f-1(U*) Í (f-1(U))* by (a).  (d) Suppose Y = V for an open effective sieve V on X. Then f-1(U*) = V Ç U* = V* Ç U* = (V Ç U)* =  (f-1(U))*.  (e) If C is neat any sieve U on X is the union of its open effective subsieves. By definition U* is the joint of open sieves on U, so U Í U*.  Suppose (C, GC) and (D, GD) are two framed sites. We consider a fixed functor F: C ® D. Suppose X is an object of C. If W is any sieve on F(X) we write FX*(W) for (FX-1(W))*. We obtain an order-preserving map FX*: wD(F(X)) ® GC(X).  Proposition 6.2. (a) If U and V are sieves on F(X), then FX*(U Ç V) = FX*(U) Ç FX*(V).  (b) For any morphism f: Y ® X in C and any sieve W on F(X) we have f-1(FX*(W)) Í FY*(F(f)-1(W)).  (c) For any open effective morphism f: Y ® X in C and any sieve W on F(X) we have f-1(FX*(W)) = FY*(F(f)-1(W)).  (d) Suppose  F: C ® D is a strict functor. Suppose X is an object of C and W an active sieve on F(X). Then FX*(W) Í FX-1(W). If C is neat then FX*(W) = FX-1(W).    Proof. (a) We have FX*(U Ç V) = (FX-1(U Ç V))* = (FX-1(U) Ç FX-1(V))* = (FX-1(U))* Ç (FX-1(V))* = FX*(U) Ç FX*(V) by (6.1.b).  (b) We have f-1(FX*(W)) Í (f-1(FX-1(W))* = (FY-1(F(f)-1(W))* = FY*(F(f)-1(W)) by (6.1.c).  (c) We have f-1(FX*(W)) = (f-1(FX-1(W))* = (FY-1(F(f)]-1(W))* = FY*(F(f)-1(W)) by (6.1.d).  (d) Suppose {Ui} is an open cover of FX*(W) such that each Ui is an open subsieve of FX-1(W). Suppose f: Y ® X is a morphism in FX*(W). Then U = f-1(È {Ui}) is the unoin of an open cover of Y. Since F is strict, F(Y) is a colimit of F(U) (i.e., F(Y) = F(U); cf. (1.3)). For each g Î U we have F(fg) Í W, thus F(fg) can be factored uniquely through the inclusion morphism W ® F(X). This implies that the morphism F(f): F(Y) ® F(X) can be factored uniquely through the inclusion W ® F(X), i.e., f Î FX-1(W). This shows that FX*(W) Í FX-1(W). If C is neat then FX-1(W) Í (FX-1(W))* = FX*(W) by (6.1.e), thus FX*(W) = FX-1(W).  Definition 6.3. We say an object X of C is geometric over D (for F) if FX* maps an open cover of F(X) to an open cover of X. We say that X is strongly geometric over D if any open subobject of X is geometric over D. A morphism f: Y ® X in C is a geometric morphism over D if f-1(FX*(W)) = FY*(F(f)-1(W)) for any open effective sieve W of F(X). We say that f is strongly geometric over D if for any open effective sieve U of X  with V = f-1(U), the restriction f|V: V ® U is geometric.  Proposition 6.4. (a) Any open subobject of a strongly geometric object over D is strongly geometric.  (b) Any open effective morphism in C is strongly geometric over D.  (c) A composition of strongly geometric morphisms of strongly geometric objects over D is a strongly geometric morphism.  Proof. (a) and (c) follow directly from the definition (6.3), and (b) is the content of (6.2.c).  It follows from (6.4) that the collection of strongly geometric objects with the strongly geometric morphisms over D forms a subsite Geo(C/F) of C, called the geometric subsite of C over D.  Lemma 6.5. Suppose U is a geometric open subobject of an object X of C. Suppose U Í FX-1(W) for an open sieve W on F(X)  and {Wi} is an open cover of W. Then {U Ç FX*(Wi)} is an open cover of U.  Proof. Applying (6.2.c) to the open effective morphism f: U ® X we get U Ç FX*(Wi) = f-1(FX*(Wi)) = FU*(F(f)-1(Wi)). Since {Wi} is an open cover of F(X), {F(f)-1(Wi)} is an open cover of F(U). But U is geometric over C, so {FU*(F(f)-1(Wi))} is an open cover of U. This implies that {U Ç FX*(Wi)} is an open cover of U.  Proposition 6.6. (a) Suppose X is strongly geometric over D. The map FX*: GC(F(X)) ® GD(X) is a morphism of frames (i.e., G(FX): GD(X) ® GC(F(X)) is a morphism of locales).  (b) Suppose X and Y are strongly geometric objects over D and f: Y ® X is a geometric morphism over D. Then f-1(FX*(W)) = FY*(F(f)-1(W)) for any open sieve W of F(X) (i.e., G(FX)G(f) = G(F(f))G(FY)).  Proof. (a) We only need to show that FX* preserves joints in view of (6.1.b). Suppose W is an open sieve on F(X) and {Wj | j Î J} is an open cover of W. Let {Ui | i Î I} be an open effective cover of FX*(W) with Ui Í FX-1(W). Since X is strongly geometric, each Ui is geometric, thus by (6.5) {Ui Ç FX*(Wj) | j Î J} is an open cover of Ui. This shows that {FX*(Wj) | j Î J} is an open cover of FX*(W).  (b) Since X and Y are strongly geometric objects over D, FX* and FY* are morphisms of frames by (a). Thus f-1FX* and FY*F(f)-1 are two morphisms of frames from G(F(X)) to G(Y). Since f is geometric over D, these two maps agree at any open effective sieve on F(X), and therefore also agree at any open sieve on F(X) because open effective sieves on F(X) form a base for G(F(X)).  Theorem 6.7. Suppose X is an object of C and {Ui | i Î I} an open effective cover of X.  (a) If each Ui is geometric (resp. strongly geometric) over D, then X is geometric (resp. strongly geometric) over D.  (b) Suppose X and Y are strongly geometric objects over D. Suppose f: Y ® X is a morphism such that the restriction f|Vi: Vi ® Ui of f on each Vi = f-1(Ui) is strongly geometric. Then f is strongly geometric over D.  Proof. (a) First suppose each Ui is geometric over D. Suppose {Wj | j Î J} is an open cover of F(X). Then by (6.5) {Ui Ç FX*(Wj) | j Î J} is an open cover of Ui. Thus {FX*(Wj) | j Î J} is an open cover of X, i.e., X is geometric over D.  Next suppose each Ui is strongly geometric. Then for any open subobject U of X there exist an open effective cover of U consisting of geometric objects over C. Applying the above result we see that U is geometric. Thus X is strongly geometric over D.  (b) Under the assumption we have G(FUi)G(f|Vi) = G(F(f|Vi))G(FVi) for each i by (6.6.b). Since G(FUi) and G(F(f|Vi)) are morphisms of locales by (6.6.a), G(FUi)G(f|Vi) and G(F(f|Vi))G(FVi) are morphisms of locales. This implies that the restrictions of the morphisms of locales G(FX)G(f) and G(F(f))G(FY) on each Vi are equal. Since {Vi} is an open effective cover of Y, we have G(FX)G(f) = G(F(f))G(FY) as Loc is strict by (5.1). This proves that f is geometric. For any open effective sieve U of X with V = f-1(U), the induced morphism f|V: V ® U satisfies the same condition, thus f|V is geometric, hence f is strongly geometric over D.  [Next Section][Content][References][Notations][Home] 