On the notion of bimodel for functorial semantics

There are various doctrines wherein theories are small categories and the models of a theory constitute a full subcategory of the category of functors from the theory to sets. We define the notion of bimodel at the nondoctrinaire level where those specified subcategories are arbitrary; as a first application we look for a classification of the equivalences between categories of models. We obtain the expected formulation under two distinct hypotheses on these subcategories of models: the case where the subcategory is reflective (the importance of which was pointed out by Pultr in [14]) is treated in Section 5 and the second case, where the subcategory contains all representable functors and has certain colimits, is treated in Section 6. In Section 7 we give relevant examples which in particular illustrate that neither of these two cases includes the other. In the last section we study in detail the case of algebraic theories.