Categories graded and bigraded by monoidal categories:
functor categories and bifunctors over non-symmetric bases
Rory Lucyshyn-Wright
Brandon University
Abstract:
Categories graded by a monoidal category V, or (V-)graded categories,
were introduced by Richard Wood (who called them large V-categories),
and they have appeared in work of several authors: Kelly, Labella,
Schmitt, and Street (who called them procategories); Garner; Levy (who
called them locally graded categories); McDermott and Uustalu. Graded
categories generalize both V-enriched categories and V-actegories while
demanding no assumptions on V. In this talk, we introduce (left)
V-graded categories and their dual notions, namely right V-graded
categories and V-cograded categories, as well as a notion of
V-W-bigraded category for a pair of monoidal categories V and W. We
discuss the speaker's recent results on functor categories and
bifunctors for graded and bigraded categories. These results enable the
consideration of graded functor categories and (bi)graded bifunctors
for arbitrary monoidal categories. Contrastingly, the familiar methods
of enriched category theory permit the consideration of enriched
functor categories and bifunctors only when the base of enrichment is
assumed symmetric, braided, or (more generally) normal duoidal in the
sense of Garner and López-Franco.