At last week’s Topos Colloquium, Rory Lucyshyn-Wright told us about categories graded by a monoidal category, following his recent preprint (Lucyshyn-Wright 2025). Graded categories, short for locally graded categories, were first introduced by Richard Wood under a different name (Wood 1976, 1978). Graded categories are of mathematical interest because they simultaneously generalize actions of a monoidal category (“actegories”) and, via a Yoneda-type embedding, enriched categories, while enjoying the advantage that extra monoidal structure like symmetry is not needed to construct functor categories and bifunctors.
This is a companion discussion topic for the original entry at https://topos.institute/blog/2025-05-01-graded-categories/