V-graded Categories for Enrichment and Actions of Monoidal Categories

This talk from the Topos Institute Colloquium on April 24, 2025, explores the concept of V-graded categories as a unifying framework for both enriched categories and categories with monoidal actions. Discover how V-graded categories provide a more general setting than traditional enriched categories, which typically require a biclosed monoidal category V that is self-enriched. Learn about V-actegories (categories equipped with an action of V) and their advantage of accommodating any monoidal category V. The presentation examines Richard Wood's approach to V-graded categories, which can be understood both as categories enriched in a monoidal category of presheaves on V and through an elementary definition involving morphisms with additional V parameters. Explore alternative terminology for these structures, including "procategories" (Kelly-Labella-Schmitt-Street) and "locally V-graded categories" (Levy), and gain insights into this mathematical framework that bridges different categorical structures.