V-graded categories and V-W-bigraded categories: Functor categories and bifunctors over non-symmetric bases

TL;DR

This paper develops a general theory of functor categories and bifunctors over non-symmetric monoidal bases using V-graded categories, extending enriched category theory to broader contexts.

Contribution

It introduces V-graded categories to unify and generalize V-enriched categories and V-actegories, enabling functorial constructions over arbitrary monoidal categories.

Findings

V-graded modules and presheaves are examples of graded bifunctors and functor categories.

The framework applies to V-enriched and V-actegory contexts, broadening categorical tools.

Comparison with existing theories is made in the case of normal duoidal categories.

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 L\'opez 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.