Localisation for cohomological theories with open–closed recollement yields a torsor of supported refinements whose secondary indeterminacy is controlled by the connecting morphism, with canonicity under Gysin and Euler injectivity.
Lurie,Derived Algebraic Geometry I: Stable Infinity Categories, arXiv:math/0608228
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A categorical and algebro-geometric theory of localization
Localisation for cohomological theories with open–closed recollement yields a torsor of supported refinements whose secondary indeterminacy is controlled by the connecting morphism, with canonicity under Gysin and Euler injectivity.