John Kennison mentioned to me that a kind of dual of the Jonsson-Tarski topos is also a topos. The JT topos has as objects pairs (X,sigma), where sigma: X --> X x X is an isomorphism and compatible functions as arrows. This topos has as objects pairs (X,tau) where tau: X + X --> X is an isomorphism and compatible functions as arrows. Note that both categories are full subcategories of the topos of M-sets where M is the free monoid on 2 generators, described by requiring a certain arrow to be an isomorphism. The JT topos is, in fact, sheaves for the topology in which separated objects are ones for which the arrow X --> X x X is injective. This Kennison topos is different since it is not sheaves for a topology. In fact, the terminal object is not 1, but rather 2^N, the structure maps given by (a_0,a_1,...) |--> (0,a_0,a_1,...) and (a_0,a_1,...) |--> (1,a_0,a_1,...) which obviously gives an isomorphism of 2^N + 2^N to 2^N. Furthermore, it is quite easy to see that the Kennison topos is boolean, so that suggests looking at not-not sheaves in the topos M-set/2^N and that is correct. If we denote the two unary ops by u and v, then any X --> 2^N already has the property that uX is disjoint from vX. Given such an X and a subobject A, the not-not closure of A consists of all x in X for which there is a word w in M with wx in A. From this, it is easy to see that a not-not separated object is one for which u and v are both injective and also easy to see that the not-not closure of the union of uX and vX (which is a subobject of X) is all of X, so that a sheaf is one for which X + X --> X is an isomorphism. I guess the "two operator" case can be replaced by any number. Another point is that though boolean, this topos is not well-pointed. For example, it is easy to find an object (X,tau) for which u has no fixed point. But this makes it impossible for there to be an arrow 2^N --> X, since (0,0,...) is fixed by u. John has asked me to add the following, which is his description of the exponential (since it is boolean, Omega = 1 + 1 and 1 is described above): I might want to add the following description of the exponential objects: Given such a coalgebra (or set X with an equivalence between X+X and X) and given c in the Cantor Set C (which is 2^{Aleph0}) we define X(c) as the fibre of the unique coalgebra homo from X to C. Then given two coalgebras A,B we will construct M = B^{A}. The elements of M are pairs (m,c) with c in C and m a function from A(c) to B(c). If f is any unary operation, then f(m,c)=(n,d) where d=f(c) and n = fmg where f is f_{B} and g is the inverse of f_{A} which exists as f_{A} must be one-to-one, and which maps to the right place. 26-Jun-2002 16:17:34 -0300,1549;000000000000-00000000