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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 ${\displaystyle S\subset A,}$ 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 unique function ${\displaystyle f|^{T}}$ 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. The term is well known in category theory, while rarely used in print.

Andreotti 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.

References

1. Dauns, John; Hofmann, Karl Heinrich (1968). Representation of rings by sections. Memoirs of the American Mathematical Society. Vol. 83. American Mathematical Society. p. ix. ISBN 978-0-8218-1283-9. MR 0247487.
2. nlab, Image, https://ncatlab.org/nlab/show/image
3. (Definition 3.1 and Remarks 3.2) in Gabriella Böhm, Hopf algebroids, in Handbook of Algebra (2008) arXiv:0805.3806
4. paragraph 2-14 at page 14 of Andreotti, A., Généralités sur les categories abéliennes (suite) Séminaire A. Grothendieck, Tome 1 (1957) Exposé no. 2, http://www.numdam.org/item/SG_1957__1__A2_0

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