categorical direct productCategoricalDirectProduct2002-04-20 01:55:192008-10-25 15:51:20Definitionproductcategorical productcategorical direct productprojection morphism% 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 \emph{direct product} of the collection $\{C_i\}_{i \in I}$ is an object $\prod_{i \in I} C_i$ of $\mathcal{C}$, with morphisms $\pi_i\colon \prod_{j \in I} C_j \to C_i$ for each $i \in I$, such that:
For every object $A$ in $\mathcal{C}$, and any collection of morphisms $f_i\colon A \to C_i$ for every $i \in I$, there exists a unique morphism $f\colon A \to \prod_{i \in I} C_i$ making the following diagram commute for all $i \in I$.
\[
\xymatrix{
A \ar@{-->}[dr]_{f} \ar[rr]^{f_i} & & C_i \\
& \prod_{j \in I} C_j \ar[ur]_{\pi_i}
}
\]
The morphisms $\pi_i\colon \prod_{j \in I} C_j \to C_i$ are called \emph{projection morphisms}.
The direct product of a finite collection of sets $C_1, C_2, \ldots, C_n$ is often denoted $C_1 \times C_2 \times \cdots \times C_n$, in analogy with the Cartesian product.