Let be a set. Frequently, is a group or monoid (usually commutative).
An -graded set is an -indexed family of sets . This can equivalently be described as a function , or as a function (a bundle with the fiber over ).
The elements of are often said to have degree .
Given a pair of -graded sets and , a homomorphism between them is an -indexed family of functions . This can equivalently be described as a natural transformation between the two associated functions , or as a function from to that make the diagram with the associated functions and commute.
The most common choices of are probably:
the natural numbers .
the integers .
the 2-element set . In this case, the elements of degree are often called even, and those of degree odd.
Suppose is a monoid, written additively. Then the category of -graded sets has a closed monoidal structure, where
This is a special case of Day convolution.
Furthermore, this monoidal structure laxly interchanges with the pointwise product of graded sets:
where the lax interchange maps are given by inclusions. These interacting monoidal structures make into a duoidal category.
A -enriched category is a category whose morphisms all have degrees in , and such that identity morphisms have degree and . Note that its underlying ordinary category, in the usual sense of enriched category theory, is the category of degree- morphisms.
More generally, we may grade by a monoidal category. This leads to the notion of locally graded category.
Given any set and any category , the category of -graded objects of is simply the functor category (identifying with its discrete category). This includes graded sets as above, as well as graded abelian groups, graded modules, graded vector spaces. However, graded rings and graded algebras are not the same (and in particular require to be a monoid).
Last revised on June 29, 2023 at 10:10:57. See the history of this page for a list of all contributions to it.