One as a relative Yoneda cohomology, i.e., as equivalence classes ofn-resolutions which are relatively split, and the second as a derived functor.
"We are a derived business," Das said.
Indeed, by its very definition, the Bousfield localization functor is the derived functor of (mathsf{Id}:mathcal {C}rightarrow mathcal {C}), and then by Quillen Adjunction Theorem, it comes equipped with an adjoint.
Moreover, we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form Aq. Finally, we show that the correspondence of unitary representations behaves well with respect to associated cycles.
Then the induced functor (Phi _cdot :C_cdot (mathcal {C}) rightarrow C_cdot (mathcal {E})) is right-Quillen, hence also right-derivable with respect to the injective model structures, and (R^cdot Phi _cdot ) is the usual derived functor of the functor (Phi ).
In general, (Phi ) is not the derived functor of any functor between the abelian categories.
Both (M^G) and (M_G) are trivial G-modules, and one defines the cohomology groups (H^i(G,M)) as the derived functors of the functor of invariants, while the homology groups (H_i(G,M)) are the derived functors of the functor of co-invariants.
Recall that for any small category I and functor E from I to abelian groups, the cohomology groups (H^cdot (I,E)) are given by begin{aligned} H^cdot (I,E) = R^cdot mathsf{lim}_I(E), end{aligned}where (R^cdot mathsf{lim}_E) are the derived functors of the limit functor (mathsf{lim}_I).
Finally, the functors (U_dagger ) and (U^dagger ) are also obviously fully faithful, and since U is both left and right-derivable, their derived functors are fully faithful as well.
