Actions, enrichments and universal enrichments

<p>This thesis studies the relationship between actions of monoidal categories and enrichments in monoidal categories, and uses these ideas to prove the existence of <em>universal enrichments</em> of a category.</p> <p>It first details the correspondence between actions of a monoidal category �� on a category �� which are right-closed, and enrichments of �� in �� which are copowered, and explains how this restricts back to an equivalence between cocomplete enrichments of the cocomplete category ��, and right-closed monoidal actions which preserve colimits in the A-variable.</p> <p>It then considers the locally presentable case, showing that any locally presentable category �� has an enrichment in the locally presentable monoidal category Cocts(��, ��) of cocontinuous endofunctors of ��. It shows that this is a <em>universal </em>cocomplete enrichment of ��, in the sense that any other cocomplete enrichment of �� is obtained from the universal one via base-change.</p> <p>It then considers the case of actions of braided monoidal categories �� on monoidal categories ��. It shows that right-closed such actions correspond to enrichments of �� to a <em>monoidal </em>��-category. Again, there is a notion of universal enrichment, and the thesis concludes by sketching a proof that a locally presentable monoidal �� has a universal monoidal enrichment in its monoidal centre ��(��).</p>