categorical direct sumCategoricalDirectSum2002-04-21 22:06:312002-04-24 10:35:35Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $\{C_i\}_{i \in I}$ be a set of objects in a category $\mathcal{C}$. A {\em direct sum} of the collection $\{C_i\}_{i \in I}$ is an object $\coprod_{i \in I} C_i$ of $\mathcal{C}$, with morphisms $\iota_i: C_i \longrightarrow \coprod_{j \in I} C_j$ for each $i \in I$, such that:
For every object $A$ in $\mathcal{C}$, and any collection of morphisms $f_i: C_i \longrightarrow A$ for every $i \in I$, there exists a unique morphism $f: \coprod_{i \in I} C_i \longrightarrow A$ making the following diagram commute for all $i \in I$.
$$
\xymatrix{
C_i \ar[dr]_{\iota_i} \ar[rr]^{f_i} & & A \\
& \coprod_{j \in I} C_j \ar@{-->}[ur]_{f}
}
$$