group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
To a permutative category $C$ is naturally associated a Gamma-space, hence a symmetric spectrum. The generalized (Eilenberg-Steenrod) cohomology theory represented by this is called the (algebraic) K-theory of (or represented by) $C$.
If the category is is even a bipermutative category then the corresponding K-theory of a bipermutative category in addition has E-infinity ring structure, hence is a multiplicative cohomology theory.
Write $FinSet^{*/}$ for the category of pointed objects finite sets.
For $C$ a permutative category, there is naturally a functor
such that (…).
(Elmendorf-Mandell, theorem 4.2)
Accordingly, postcomposition with the nerve $N : Cat \to sSet$ produces from $C$ a Gamma-space $N \widebar C$. To this corresponds a spectrum
This is the K-theory spectrum of $C$.
Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)
Anthony Elmendorf, Michael Mandell, Permutative categories as a model of connective stable homotopy, in: Birgit Richter (ed.) Structured Ring spectra, Cambridge University Press (2004)
Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, K-Theory 0680 (web, pdf)