Definition:Product Category
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Definition
Let $\mathbf C$ and $\mathbf D$ be metacategories.
The product category $\mathbf C \times \mathbf D$ is the category with:
Objects: | $\tuple {X, Y}$, for all $X \in \operatorname {ob} \mathbf C$, $Y \in \operatorname {ob} \mathbf D$ | |
Morphisms: | $\tuple {f, g}: \tuple {X, Y} \to \tuple {X', Y'}$ for all $f: X \to X'$ in $\mathbf C_1$ and $g: Y \to Y'$ in $\mathbf D_1$ | |
Composition: | $\tuple {f, g} \circ \tuple {h, k} := \tuple {f \circ h, g \circ k}$, whenever this is defined | |
Identity morphisms: | $\operatorname {id}_{\tuple {X, Y} } := \tuple {\operatorname {id}_X, \operatorname {id}_Y}$ |
Also see
- Results about product categories can be found here.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.1$