On exponentiating exponentiation

Abstract

It is well known that for any (small) category J, the category EnsJ of functors from J into the category of sets, is Cartesian closed (in fact it is an elementary topos, see for example Lawvere [l] or Tierney [4]). An exercise in MacLane [3, p. 961 suggests that a functor category KJ is Cartesian closed whenever K is Cartesian closed; however, it is not difficult to find isolated counter-examples to this (as indicated by changes in the exercise in later editions of the book). The motivation for this paper was to give a systematic treatment of those situations in which KJ inherits from K the property of Cartesian closedness. The main results are the following: IfK is complete and Cartesian closed then KJ is Cartesian closedfor all J; moreover, exponentiation in KJ is explicitly constructed by forming certain limits in K. Call a category J right-finite whenever, for each object A of J, there are, up to isomorphism under A, only finitely many morphisms with domain A. If J is right-finite then KJ is Cartesian closed for all Cartesian closed K; this is proved by showing that for such J the above-mentioned limits are essentially finite. For the case that J is a partially ordered set, right-finiteness is both a necessary as well as a sufficient condition for KJ to be Cartesian closed for all Cartesian closed K. This paper is divided into three sections. The first deals with the case that both J and K are partially ordered sets; here, right-finiteness for J means just that every principal filter is finite, and K being Cartesian closed means that it is a relatively pseudo-complemented meet-semilattice. A very simple argument is pr~esented showing that for such K, if either J is right-finite or K is complete then KJ is Cartesian closed. Further, it is proved that if J is not right-finite then for every countably incomplete Boolean algebra B, BJ is not Cartesian closed. Section two is devoted to proving the main result (Proposition 3); as an application we see that for