A generalization of Gabrielʼs Galois covering functors and derived equivalences

A Generalization of Gabriel’s Galois Covering Functors II: 2-Categorical Cohen-Montgomery Duality

“…Here we modified the definition of degree-preserving functors in order to include the functor H (and hence the functors ω ′ B for all G-graded categories B, see Definition 8.7) in Proposition 6.4 below because H is not degree-preserving in the sense of [1] in general (see [1,Remark 5.9] and Remark 8.9). …”

“…Let (F, φ) : C → B be a G-invariant functor and x, y ∈ C. Then we define homomorphisms F (1) x,y := (F, φ) (1) x,y and F (2) x,y := (F, φ) (2) x,y of k-modules as follows.…”

“…For instance (a) was investigated in [4], [11], and (b) was examined in [6], [1]. To be more precise let C be a category with a G-action and B a G-graded category.…”

“…(i) Are the G-equivariant equivalence ε C and the degree-preserving equivalence ω B obtained in [1] equivalences defined by 2-categorical structures on G-Cat and G-GrCat, respectively?…”

“…In sections 2 and 3 we define the 2-category G-Cat and the 2-category G-GrCat, respectively. In sections 4, 5 and 6 we recall from [1] fundamental facts about G-covering, the definition and characterizations of orbit categories, and fundamental facts about smash products, respectively. In section 7 we extend the orbit category construction and the smash product construction to 2-functors ?/G and ?#G, respectively, and give the precise statement of the main result.…”