Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let \(\mathbb{k}\) be a commutative ring and I a category. As a generalization of a \(\mathbb{k}\)-category with a (pseudo) action of a group  consider a family of \(\mathbb{k}\)-categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small \(\mathbb{k}\)-categories.  investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.
['Hideto Asashiba']
I am writing a research paper on a idea Derived Equivalences of Actions of a Category
[{"from": "human", "value": "I am writing a research paper on a idea Derived Equivalences of Actions of a Category"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let \\(\\mathbb{k}\\) be a commutative ring and I a category. As a generalization of a \\(\\mathbb{k}\\)-category with a (pseudo) action of a group  consider a family of \\(\\mathbb{k}\\)-categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small \\(\\mathbb{k}\\)-categories.  investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors."}]