In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual.
Given any subset formula_1 we can consider the corresponding inclusion of sets formula_2 as a function. Then for any function formula_3, the restriction formula_4 of a function formula_5 onto formula_6 can be defined as the composition formula_7.
Analogously, for an inclusion formula_8 the corestriction formula_9 of formula_5 onto formula_11 is the unique
function formula_12 such that there is a decomposition formula_13. The corestriction exists if and only if formula_11 contains the image of formula_5. In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of formula_5. More generally, one can consider corestriction of a morphism in general categories with images. The term is well known in category theory, while rarely used in print.
Andreotti introduces the above notion under the name , while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if formula_17 is a surjection of sets (that is a quotient map) then Andreotti considers the composition formula_18, which surely always exists.
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