Английская Википедия:Corestriction

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In mathematics, a corestrictionШаблон:Better source needed 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 <math>S\subset A,</math> we can consider the corresponding inclusion of sets <math>i_S:S\hookrightarrow A</math> as a function. Then for any function <math>f:A\to B</math>, the restriction <math>f|_S:S\to B</math> of a function <math>f</math> onto <math>S</math> can be defined as the composition <math>f|_S = f\circ i_S</math>.

Analogously, for an inclusion <math>i_T:T\hookrightarrow B</math> the corestriction <math>f|^T:A\to T</math> of <math>f</math> onto <math>T</math> is the unique function <math>f|^T</math> such that there is a decomposition <math>f = i_T\circ f|^T</math>. The corestriction exists if and only if <math>T</math> contains the image of <math>f</math>. In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of <math>f</math>. 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 Шаблон:Lang, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if <math>p^U:B\to U</math> is a surjection of sets (that is a quotient map) then Andreotti considers the composition <math>p^U\circ f:A\to U</math>, which surely always exists.