| Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$. | |
| A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$. | |
| Definition: An initial morphism $f:x\to y$ such that the underlying morphism $U(f):U(x)\to U(y)$ is monic is called an $\textbf{embedding}$. | |
| Definition: In a concrete category an object $I$ is called $\textbf{injective}$ provided that for any embedding $m: A \to B$ and any morphism $f: A\to C$ there exists a morphism $g:B\to C$ extending $f$, i.e., $g\circ m=f$ | |
| Theorem: In $\mathcal{A}\mathrm{b}$ the injective objects are precisely the divisible abelian groups. |