Manufacturing a cartesian closed category with exactly two objects out of a C-monoid

A construction is described of a cartesian closed category . with exactly two elements out of a C-monoid that 9.can be recovered from A without reference to the construction. Note: The first author was partially supported by the Dutch government through the SPIN project PRISMA; the second author was partially supported by the EEC through Esprit project 415. We answer a question of Lambek and Scott (see [LS] p.99) by proving the following: Theorem. Let 9vt be a C-monoid, with C-structure (7t, 7t', £, (_)*, <_,_>). Then there exists a cartesian closed category A with exactly two objects U and T, such that End(U) = !M The construction of . is entirely by hand. The intuitive idea is as follows. !may be viewed as a collection of endomorphisms of a set U. Let T_ { *I be a one-point set; then u X*.u is a one-to-one correspondence between U and the set of all functions from T to U. Now if . is a cartesian closed category with just U and T for its objects, where T is terminal, then in A we must have Hom(U,U) = Hom(TxU,U) = Hom(T,UU) =_ Hom(T,U); so if we put Hom(U,U) = iM, and like to think of Hom(T,U) as HomSets({*},U), we must have M_ U, as sets. Since it does not matter much what the elements of U are, we take M=U. Then we have functions ft _ k*.f : *I U for every f E U. Composing with o ku.*:U-{*}, we have (X*.f) o (ku.*) = ku f : UU. This we identify with the arrow Xuf _ (fn')* in ?tt, described in [LS] §15. The longer definitions (notably, those of goft and {gt,ht}) were forced upon us by this identification. The rest were the simplest at first sight. Remark. By [LS] § 16, the Karoubi envelope K(it) of has a full cartesian closed subcategory Kp(" consisting of all objects isomorphic to U (the unit of iM) or the terminal object T. Taking one representative from either isomorphism class, one gets another full subcategory,