Full continuous embeddings of toposes

Abstract

Some years ago, G. Reyes and the author described a theory relating first order logic and (Grothendieck) toposes. This theory, together with standard results and methods of model theory, is applied in the present paper to give positive and negative results concerning the existence of certain kinds of embeddings of toposes. A new class, that of prime-generated toposes is introduced; this class includes M. Barr's regular epimorphism sheaf toposes as well as the so-called atomic toposes introduced by M. Barr and R. Diaconescu. The main result of the paper says that every coherent prime-generated topos can be fully and continuously embedded in a functor category. This result generalizes M. Barr's full exact embedding theorem. The proof, even when specialized to Barr's context, is essentially different from Barr's original proof. A simplified and sharpened form of Barr's proof of his theorem is also described. An example due to J. Malitz is adapted to show that a connected atomic topos may have no points at all; this shows that some coherence assumption in our main result is essential. Introduction. This paper has three sources. One is Michael Barr's full exact embedding theorem [2] for exact categories, generalizing the Lubkin-FreydMitchell embedding theorems for Abelian categories. Another is the theory of (Grothendieck) toposes, especially coherent toposes (see SGA4 [1]) and the third is model theory (see CK [4]). We have found a particular kind of topos, the so-called prime-generated ones that comprise both the regular epimorphism sheaf toposes related to Barr's regular categories and atomic toposes introduced in Barr and Diaconescu [3] (also considered by A. Joyal). We prove a full embedding theorem for coherent prime-generated toposes that generalizes Barr's theorem and specializes in a new result for coherent atomic toposes. Our chief tools are model theoretical; we use the mechanisms introduced in [10] relating categories and logic. Specifically, we call an object A of a topos & a prime object if it cannot be written as a supremum of proper subobjects. & is a prime-generated topos, if the prime objects in & form a collection of generators for 6. For toposes &, &' a functor & -> 6' is continuous if it is left exact and has a right adjoint, i.e. if it is the inverse-image functor of a geometric morphism &' -> 6. Our main results are that every prime-generated coherent topos admits a full continuous embedding into a functor category (K, SET) with a small category K (Corollary 2.7) and that, in fact, Received by the editors June 4, 1979. 1980 Mathematics Subject Classification. Primary 03G30, 18B15; Secondary 03C50.