V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases
Rory B. B. Lucyshyn-Wright
TL;DR
The paper generalizes functor categories and bifunctor calculus to arbitrary monoidal bases by developing a theory of V-graded categories, left and right gradings, and bigraded structures that subsume both V-enriched categories and V-actegories. It introduces graded functor categories valued in bigraded targets and graded bifunctors, along with a bigraded product and an enveloping actegory to formalize graded diagrams. Core results include representations of graded bifunctors by graded functor categories, Yoneda-type correspondences, and a cohesive treatment of non-symmetric bases via contravariant base change, Day convolution, and (duoidal) Garner–López Franco comparisons. The framework unifies and extends classical enrichment theory beyond symmetry assumptions, providing explicit constructions for V-graded modules, presheaves, and their Yoneda theory, with broad implications for category-theoretic foundations and potential applications in computer science and related fields.
Abstract
In the well-known settings of category theory enriched in a monoidal category V, the use of V-enriched functor categories and bifunctors demands that V be equipped with a symmetry, braiding, or duoidal structure. In this paper, we establish a theory of functor categories and bifunctors that is applicable relative to an arbitrary monoidal category V and applies both to V-enriched categories and also to V-actegories. We accomplish this by working in the setting of (V-)graded categories, which generalize both V-enriched categories and V-actegories and were introduced by Wood under the name "large V-categories". We develop a general framework for graded functor categories and graded bifunctors taking values in bigraded categories, noting that V itself is canonically bigraded. We show that V-graded modules (or profunctors) are examples of graded bifunctors and that V-graded presheaf categories are examples of V-graded functor categories. In the special case where V is normal duoidal, we compare the above graded concepts with the enriched bifunctors and functor categories of Garner and López Franco. Along the way, we study several foundational aspects of graded categories, including a contravariant change of base process for graded categories and a formalism of commutative diagrams in graded categories that arises by freely embedding each V-graded category into a V-actegory.