Graded categories as double functors

Evan Patterson

Fri May 02 2025

I explain how a category graded by a monoidal category can be viewed as a double functor out of the delooping of the monoidal category. A few consequences and a series of examples are then presented.

This is a companion discussion topic for the original entry at https://www.epatters.org/post/graded-categories

Eigil Rischel

Fri May 02 2025

A very important class of examples are the categories of parameterized morphisms, that is taking \mathcal{C}_m(x,y) = \operatorname{Hom}(m \cdot x, y) for some action of the monoidal category \mathcal{M}. In fact I think this should give a 2-equivalence of category actions with a subcategory of graded categories (the “representable” ones), but I’m not aware of this being written down anywhere.

varkor

Fri May 02 2025

While it’s not stated in exactly this language, this perspective on actions of a monoidal category appears in Wood’s thesis on graded categories (see pages 41 – 43 of Wood’s “Indicial methods for relative categories”). For a statement in more modern language, see for instance Proposition 4.14 of Campbell’s “Skew-enriched categories”.

Evan Patterson

Fri May 02 2025

Thanks both for the comments! Eigil, also have a look at Example 3.9 in Lucyshyn-Wright’s paper, where V-actegories are shown to be 2-equivalent to the full sub-2-category of V-graded categories spanned by the V-graded categories with V-copowers. Lucyshyn-Wright points out that this goes back to Wood, as Nathanael says.