On compact space objects in topoi

Abstract

As E. Manes [3] has shown, the se t based category of compact Hausdorff spaces and their continuous maps is isomorphic to the category of algebras for the ultrafilter monad on sets. Several proofs of this result have been given; they show clearly that the isomorphism between the categories of compact Hausdorff spaces and of ultrafilter monad algebras is closely tied in with the Axiom of Choice. Thus the question is legitimate: what happens if we replace by a topos? When trying to answer this question, we are immediately confronted with another question: how do we generalize ultrafilters? Should we use prime filters, or should we use uitrafilters as defined by H. Volger [8]? Should we replace non -emp ty by n o n initial or by inhabited? We have not been able to decide between the various possibilities; thus we shall use them on an equal footing. Our first task is to construct suitable objects and monads of prime filters and of ultrafilters; this is accomplished in Sections 2 and 3. In Section 2, we obtain submonads of the double powerset monad from propositional connectives. This allows us to construct a filter monad and a prime filter monad on a topos, as well as arithmetic lattices and Stone spaces based on a topos. In Section 3, we construct a sets of inhabited subsets monad from a contravariant adjunction. One part of this adjunction is a contravariant powerset functor, from the category of partial morphisms in a topos to the topos. One can easily show, following the method of Par~ [5], that this functor is monadic; we shall not do this here. Combining the results of these two sections, by taking intersections of submonads of the double powerset monad, we obtain all the prime filter and ultrafilter monads we may want. The algebraic side of the problem is thus in good shape; this cannot be said for the topological side which we discuss in Section 4. In this section, we define compact Hausdorff spaces in a topos, relative to a submonad 9 of the filter monad, and we obtain an induced algebra functor Jr, from compact Hausdorff spaces to T algebras. In the other direction, we construct an induced topology functor, from T algebras to topological spaces. Induced topologies of T-algebras need not be Hausdorff. If we restrict ourselves to algebras for which the induced topology is Hausdorff, then induced topologies define a right inverse left adjoint functor of the induced algebra functor J.. We use standard notations as much as possible. Additional notations and auxiliary results are collected in Section 1. In particular, we define characteristic functions of relations, and discuss some of their properties. This is one useful tool for the present paper; the Mitchell B~nabou Osius language is another. We use this language in the form presented by G. Osius in [4]. Most of the material presented in this paper is taken from the unpublished thesis of the f i rs t -named author [6].