Protomodular aspect of the dual of a topos
The notion of protomodular category has been extensively investigated (principally by the author of this paper) in recent years, as a category having enough of the `intrinsic' algebraic structure of the category of groups to permit the development of much of the machinery of nonabelian homological algebra. It was first observed by the reviewer that the category of Heyting algebras is protomodular; indeed, it provides the best known example of a protomodular variety which does not admit a weakly associative Mal'cev operation. Since the contravariant power-object functor on any topos \(\mathcal E\) can be regarded as taking values in the category of internal Heyting algebras in \(\mathcal E\), it follows easily that the dual of any topos is protomodular (as well as being Barr-exact and congruence distributive). In the present paper, the author investigates the significance of the known results about Barr-exact protomodular categories as applied to the duals of toposes, and shows that they include several familiar exactness properties of toposes.
scientific article; zbMATH DE number 2093010