Morita equivalence of sketches
This paper deals with arbitrary small sketches and their categories of models. Two sketches \({\mathcal S}\) and \({\mathcal T}\) are called Morita equivalent if their categories of models in \({\mathcal S}et\), \({\mathcal M}od({\mathcal S})\) and \({\mathcal M}od({\mathcal T})\), are equivalent. Sketches are classified by doctrines: a doctrine \({\mathcal D}\) is a pair \(({\mathcal D}_L,{\mathcal D}_C)\) of classes of small categories, a \({\mathcal D}\)-sketch is a sketch whose cones have base in \({\mathcal D}_L\) and co-cones have base in \({\mathcal D}_C\). To any \({\mathcal D}\)-sketch \({\mathcal S}\) is associated the legitimate category \(\widehat{\mathcal S}\) of those accessible functors \({\mathcal M}od({\mathcal S})\to{\mathcal S}et\) which preserve all \(D\)-limits where \(D\in {\mathcal D}_L\) is such that, in \({\mathcal S}et\), \(D\)-limits commute with all \({\mathcal D}_C\)-colimits. An \({\mathcal S}-{\mathcal T}\) bimodel is a model of \({\mathcal S}\) in \(\widehat{\mathcal T}\). Bimodels can be composed in an associative way. The main result of the paper is that two sketches \({\mathcal S}\) and \({\mathcal T}\) are Morita equivalent if and only if there exists a \({\mathcal S}-{\mathcal T}\) bimodel \(\varphi\) and an \({\mathcal T}-{\mathcal S}\) bimodel \(\psi\) which are mutually inverse for the composition. Several particular cases are studied.
scientific article; zbMATH DE number 1563364