Enriched categories are ubiquitous in mathematics. In this talk, I will introduce the basic idea of what enrichment in a (virtual) monoidal category means, and how a (virtual) bicategory and a (virtual) double category offer us natural generalizations of it. I will show many examples of enrichment in these three settings, and one construction of a new enrichment from an old one, namely the lax slice. In addition, I will explain how we used this construction to recover and enhance the result of Cockett and Garner on restriction categories as enriched categories.
This talk is based on joint work in progress with Nathanael Arkor, Yorgo Chamoun, Bryce Clarke, Vít Jelínek, and Theofilos Tsantilas, which started as the 2026 Adjoint School research project “Double Category Theoretic Perspective on Partiality”.