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.

Andreottiparagraph 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 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

{{reflist

Set theory
Functions and mappings
Category theory
Hopf algebras
Abelian group theory