Corestriction

In mathematics, a corestriction^[1] 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 ${\displaystyle S\subset A,}$ we can consider the corresponding inclusion of sets ${\displaystyle i_{S}:S\hookrightarrow A}$ as a function. Then for any function ${\displaystyle f:A\to B}$, the restriction ${\displaystyle f|_{S}:S\to B}$ of a function ${\displaystyle f}$ onto ${\displaystyle S}$ can be defined as the composition ${\displaystyle f|_{S}=f\circ i_{S}}$.

Analogously, for an inclusion ${\displaystyle i_{T}:T\hookrightarrow B}$ the corestriction ${\displaystyle f|^{T}:A\to T}$ of ${\displaystyle f}$ onto ${\displaystyle T}$ is the uniquefunction ${\displaystyle f|^{T}}$ such that there is a decomposition ${\displaystyle f=i_{T}\circ f|^{T}}$. The corestriction exists if and only if ${\displaystyle T}$ contains the image of ${\displaystyle f}$. In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of ${\displaystyle f}$. More generally, one can consider corestriction of a morphism in general categories with images.^[2] The term is well known in category theory, while rarely used in print.^[3]

Andreotti^[4] introduces the above notion under the name coastriction, while the name corestriction reserves to the notion categorically dual to the notion of a restriction. Namely, if ${\displaystyle p^{U}:B\to U}$ is a surjection of sets (that is a quotient map) then Andreotti considers the composition ${\displaystyle p^{U}\circ f:A\to U}$, which surely always exists.