FORCING AND CLASSIFYING TOPOI 9
For a geometric theory T and a Grothendieck topos F , we let
Mod(F,T) be the category of models of T in F and homomorphisms. For
a Grothendieck topos E , we often write Mod(F,E) for the category of
geometric morphisms F •+ E and natural transformation n: f* -* • g* . E
is a classifying topos of T iff Mod(F,T) ^ Mod(F,E) for all Grothen-
dieck topoi F . A geometric morphism ¥ + £ (with its inverse image f*
in mind) is often called a F-model of (a geometric theory classified by)
E . Indeed, for every Grothendieck topos E there exists a geometric
theory whose classifying topos is E .
0.8. Flat functors. Diaconescu's Theorem.
We conclude the preliminaries by stating a most important and useful
theorem of Diaconescu ([D],[Jo 1 ,§4.3.]) on characterization of geometric
morphisms into S in terms of functors on ( E , and some related lemmas.
Let F be a functor on a category ( E (to Sets). We say that F is
flat if the following conditions are met:
(1) given finitely many (possibly zero) objects A. of ( E and
elements a. e F(A.) , there is an object B of ( E , morphisms
B • A. of ( E , and an element b e F(B) such that
F(ai)(b) = ai for all i ,
(2) given a e F(A) and finitely many morphisms A - A'
of ( E such that all of the F(ct.)(a) are equal, we have
a morphism B -*• A of ( E such that all of the composites
B -* A' are equal, and a is in the image of F($) .
Notice that (1) and (2) imply that given finitely many morphisms A. - A'
of E , and elements a. e F(A.) such that all of the F(a.)(a.) are
equal, there is an object B of ( E , morphisms B -* Ai of ( E , and an
element b e F(B) such that all of the composites
ar e equal and
F(Bi) (b) = a± for all i .
It is also easy to see that a flat functor preserves finite limits.
Theorem (Diaconescu). Let ( E be a small category with a set of objects.
Let F be a Grothendieck topos. Then there is an equivalence between