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# Exercise 85 - Chapter 3

edited June 2018
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?

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