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CT (Category theory)?, AG (Algebraic geometry)?
See Deligne: Hodge III, Section 6.3.
Cat
For definition of relative sheaf cohomology, see page 249 of Iversen.
Work out relative cohomology, cohom of pairs, cohom with support etc. See Gilllet-Soule for some of this, including one possible meaning of purity. In Feliu intro, DB cohom with supp in Z is defined by the simple of the map from cohom of X to the cohom of U = X\Z.
Construction of K-theory with compact support (and also the construction of relative cohomology in general) as the homotopy fiber of a map of spectra (complexes) (I think, see D-C section 12.4 with ref to Gillet I think).
There is something in Deninger-Scholl on relative mixed motives, or is this in the Motives vol???
nLab on relative cohomology.
http://mathoverflow.net/questions/12676/relative-version-of-sheaf-cohomology
http://mathoverflow.net/questions/111059/relative-de-rham-cohomologies
nLab page on Relative cohomology