Model-theoretic Imaginaries and Coherent Sheaves

We show the equivalence of categories of model-theoretic imaginaries (of various kinds) with categories of “small” (finitely generated, finitely presented, coherent) functors. We do this first for certain locally finitely presented categories and then, by localising, for much more general “definable categories” (categories of models of coherent theories). We also investigate the corresponding notion of interpretation.