On the notion of bimodel for functorial semantics (Q1332449)

Having in mind the use of bimodules to establish a criterion for the equivalence of categories of modules (the Morita Theorem), the authors define a general notion of bimodel from the point of view of functorial semantics. Here a category of models is any full subcategory \(T_ 1\) of the category \(T^*\) of all functors from \(T^{op}\) to \textit{Set}, \(T\) being small. Giving two such categories \(T_ 1\) and \(S_ 1\), a functor \(\varphi' : T^{op} \to S_ 1\) is a \(S-T\) bimodel if the functor \(\Hom (\varphi,-) : S^* \to T^*\) defined by \(\Hom (\varphi,K) (T) = \text{Nat} (\varphi (T),K)\) restricts to a functor from \(S_ 1\) to \(T_ 1\). In the cases where the categories of models \(T_ 1\) are either reflexive in their corresponding \(T^*\), or the Yoneda functors \(T^{op} \to T^*\) factorize through \(T_ 1\) and \(T_ 1\) has ``enough'' colimits, the authors define a composition of bifunctors in such a way that \(T_ 1\) and \(S_ 1\) are equivalent if and only if there exist \(\varphi' : T^{op} \to S_ 1\) and \(\psi' : S^{op} \to T_ 1\) such that \(\varphi'* \psi'\) and \(\psi'* \varphi'\) are identities. Examples are given, as well as connections and comparisons with other concepts and results, like sketches, flat functors, Cauchy-equivalences, Morita-equivalences, distributors. The case of algebraic theories is treated in some details.