\begin{definition}[Definition:Free Abelian Group on Set]
Let $\Z$ be the additive group of integers.
Let $S$ be a set.
The '''free abelian group on $S$''' is the pair $(\Z^{(S)}, \iota)$ where:
* $\Z^{(S)}$ is the direct sum of $S$ copies of $\Z$. That is, of the indexed family $S \to \{\Z\}$
* $\iota : S \to \Z^{(S)}$ is the '''canonical mapping''', which sends $s$ to the mapping $\delta_{st} \in \Z^{(S)}$, where $\delta$ denotes Kronecker delta.
\end{definition}