nLab restriction

Definition

The restriction of a function $f: X \to Y$ to a subset $U$ of $X$ is simply the composite of $f$ and the inclusion function of $U$:

$$U \stackrel{i_U}\to X \stackrel{f}\to Y .$$

It is often written as $f_{|U}: U \to Y$, or $f|_U$, or a variation.

More generally, in any category $C$, a monomorphism $i_U : U\hookrightarrow X$, and a morphism $f \colon X\to Y$, the restriction $f|_U \colon U \to Y$ of $f$ onto $U$ is the precomposition $f|_U \coloneqq f \circ i_U$ of $f$ by $i_U$. A subobject is an equivalence class of monomorphisms. For a different representative of the subobject, $i_{\tilde{U}} \colon \tilde{U}\to X$ there is a unique isomorphism $b \colon U\to\tilde{U}$ such that $i_{\tilde{U}}\circ b = i_U$, hence $f_{\tilde{U}} = f\circ b$.

Related concepts

• extension

• lift

• Cauchy principal value