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Exercise 85 - Chapter 3

  1. What are the identity morphisms in a product category \( \mathcal{C} \times \mathcal{D} \)?
  2. Why is composition in a product category associative?
  3. What is the product category \( \textbf{1} \times \textbf{2} \)?
  4. What is the product category \( P \times Q \) when \(P\) and \(Q\) are preorders?

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Comments

  • 1.
    1. The products of identity morphism for pairs of objects, one from \(\mathcal {C}\) and the other from \(\mathcal {D}\).

    2. Because composition in each multiplicand is associative.

    3. It has objects \((1,a)\), \((1,b)\), and a single non-identity morphism: \((\mathrm{id}_1,f):(1,a)\to(1,b)\).

    4. It is the product preorder on the Cartesian product of the underlying sets (since \((f,g):(c,d)\to(c',d')\) exists iff there is a morphism from \(c\) to \(c'\) and a morphism from \(d\) to \(d'\), i.e., products are ordered iff both their multiplicands are (as in Example 1.47)).

    Comment Source:1. The products of identity morphism for pairs of objects, one from \\(\mathcal {C}\\) and the other from \\(\mathcal {D}\\). 2. Because composition in each multiplicand is associative. 3. It has objects \\((1,a)\\), \\((1,b)\\), and a single non-identity morphism: \\((\mathrm{id}_1,f):(1,a)\to(1,b)\\). 4. It is the product preorder on the Cartesian product of the underlying sets (since \\((f,g):(c,d)\to(c',d')\\) exists iff there is a morphism from \\(c\\) to \\(c'\\) and a morphism from \\(d\\) to \\(d'\\), i.e., products are ordered iff both their multiplicands are (as in Example 1.47)).
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