Corestriction

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 S\subset A, we can consider the corresponding inclusion of sets i_S:S\hookrightarrow A as a function. Then for any function f:A\to B , the restriction f|_S:S\to B of a function f onto S can be defined as the composition f|_S = f\circ i_S . Analogously, for an inclusion i_T:T\hookrightarrow B the corestriction f|^T:A\to T of f onto T is the unique function f|^T such that there is a decomposition f = i_T\circ f|^T . The corestriction exists if and only if T contains the image of f . In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of f . 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.…

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 S\subset A, we can consider the corresponding inclusion of sets i_S:S\hookrightarrow A as a function. Then for any function f:A\to B , the restriction f|_S:S\to B of a function f onto S can be defined as the composition f|_S = f\circ i_S . Analogously, for an inclusion i_T:T\hookrightarrow B the corestriction f|^T:A\to T of f onto T is the unique function f|^T such that there is a decomposition f = i_T\circ f|^T . The corestriction exists if and only if T contains the image of f . In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of f . 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 p^U:B\to U is a surjection of sets (that is a quotient map) then Andreotti considers the composition p^U\circ f:A\to U , which surely always exists. ==References==

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