On completeness of locally-internal categories

Abstract

Locally-internal categories over a topos E are regarded as categories enriched in the bicategory Span E. In this paper we develop some aspects of completeness. For locally-internal categories, completeness means: finite completeness over each fiber, transition functors along the maps of the base topōs, and a Beck-Chevalley condition (see Bénabou, C.R. Acad. Sci. Paris 281 (1975) A897–900). We prove that this notion can be obtained by particularizing to Span E the general notion of completeness of enriched category theory, given in terms of indexed limits. We give also an adjoint functor theorem.