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Definition 4.2.20. Let $\mathcal{A}$, $\mathcal{B}$ be categories. We define the product category $\mathcal{A} \times \mathcal{B}$ to be the category with objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A} \times \mathcal{B}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{A} \times \mathcal{B}}((x, y), (x', y')) := \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, x')\times \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(y, y'). \]

Composition is defined componentwise.

Comments (2)

Comment #145 by on

Delete either the first or the second occurence of "to be the category".

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