Molecular Toposes*

In [2], Barr and Diaconescu characterized those Grothendieck toposes 8 for which the inverse image, A, of the geometric morphism r: 8 + Yet, is logical. It was shown (among other things) that this happens precisely when the lattice of subobjects of every object of 8 is a complete atomic boolean algebra. Toposes satisfying this property are called atomic. These results were relativised to the case where f : 8 + Y is an arbitrary morphism of elementary toposes. Their proofs used Mikkelsen’s theorem [4] which says that a logical functor between toposes has a left adjoint if and only if it has a right adjoint, in order to obtain a left adjoint A to A. (E.g. in the Yet based case, AA is the set of atomic subobjects of A.) The purpose of this paper is to obtain analogous theorems characterizing those Grothendieck toposes 8 for which A has a left adjoint. For reasons which will become clear later, these toposes are called molecular. It is an exercise in [7, p. 414, Ex 7.61 that Sh(X) is molecular if and only if X is locally connected. We also treat the relative case, where Yeet is replaced by an arbitrary elementary topos 9 These results may be taken as a definition and characterizations of what it means for an elementary topos to be locally connected over another topos. It is presumably because of topological considerations such as these that Joyal raised the question that resulted in this paper. Tierney has also shown that our conditions are closely related to the problem of determining when a pullback of elementary toposes satisfies the Beck condition.