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