Notions of logic internal to a monoidal category

In this talk, based on joint work with J. Rosicky, I will explain how one can treat equational and regular (i.e. positive existential) logic internally to a given (and sufficiently nice) monoidal category V.  Our motivation for this comes from enriched category theory, where theories internal to V are used to classify an important class of categories enriched over V. For the purposes of this talk, however, we will deal only with the internal aspects of the logic and with examples, focusing particularly on the cases where V is taken to be the category of Abelian groups, of Posets, of Metric spaces, and of Banach spaces.