Spectral Mackey functors and equivariant algebraic \(K\)-theory. I. (Q329504)

The author defines a \textit{spectral Mackey functor} as (roughly) a family of spectra equipped with operations that mirror the operations found in ordinary Mackey functors, together with all of the homotopies and higher coherences among these operations.NEWLINENEWLINEThe key results relate spectral Mackey functors to algebraic \(K\)-theory. In one direction, the author shows that representable Mackey functors can be realized as equivariant algebraic \(K\)-theory spectra. In the other direction, he shows that the algebraic \(K\)-theories of families of Waldhausen \(\infty\)-categories, connected by suitable adjoint pairs of functors, define a spectral Mackey functor. This allows for a complete accounting of all the functorialities enjoyed by such families of Waldhausen categories.NEWLINENEWLINEThe general theory is illustrated by four examples worked out in detail and presented in four separate appendices. The last of these presents a fully functorial version of the algebraic \(D\)-theory of derived stacks and a general construction of \(\pi_1^{et}\)-equivariant algebraic \(K\)-theory for étale fundamental groups.