Injective object

Remarks

Equivalently, the contravariant hom functor $\Hom(-, I)$ sends monomorphisms to surjections of sets. In an abelian category, $I$ is injective if and only if $\Hom(-, I)$ is exact, and if and only if every short exact sequence $0 \to I \to B \to C \to 0$ splits. In the category of modules over a ring, the injective objects are the injective modules. An injective object of $\mathcal{C}$ is the same as a projective object of the opposite category.

Example: $\Q$ is injective in the category of abelian groups. It is a divisible group, and by Baer's criterion a divisible abelian group is injective: a map $f \colon n\Z \to \Q$ extends to $\Z$ by sending $1$ to $f(n)/n$.

Non-example: $\Z$ is not injective in the category of abelian groups. For $m \colon \Z \to \Z$, $m(x) = 2x$, and $f = 1_{\Z}$, an extension $g$ would need $2 g(1) = g(m(1)) = f(1) = 1$, which has no solution in $\Z$.