local object, local morphism
Bousfield localization, simplicial localization
descent object, descent morphism
sheaf, (2,1)-sheaf/stack, 2-sheaf,(∞,1)-sheaf/∞-stack
monadic descent, higher monadic descent
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A localization is called reflective if the full subcategory of local objects is a reflective subcategory/reflective sub-(∞,1)-category.