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# Exercise 72 - Chapter 2

edited June 2018

Let $$\mathcal{X} \times \mathcal{Y}$$ be the $$\mathcal{V}$$-product as in Definition 2.51.

1. Check that for every object $$(x, y) \in Ob(\mathcal{X} \times \mathcal{Y})$$ we have $$I \le (\mathcal{X} \times \mathcal{Y})((x, y), (x, y))$$.

2. Check that for every three objects $$(x_1 , y_1 ), (x_2 , y_2 ), \text{ and } (x_3 , y_3 )$$, we have $$(\mathcal{X} \times \mathcal{Y})((x_1 , y_1 ), (x_2 , y_2 )) \otimes (\mathcal{X} \times \mathcal{Y})((x_2 , y_2 ), (x_3 , y_3 )) \le (\mathcal{X} \times \mathcal{Y})((x_1 , y_1 ), (x_3 , y_3 )) \\).$$

3. We said at the start of Section 2.3.1 that the symmetry of $$\mathcal{V}$$ (condition (d) of Definition 2.1) would be required here. Point out exactly where that condition is used.

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1.
edited August 2018

Definition 2.71:

Let $$\mathcal{X}$$ and $$\mathcal{Y}$$ be $$\mathcal{V}$$-categories. Define their $$\mathcal{V}$$-product, or simply product, to be the $$\mathcal{V}$$-category $$\mathcal{X}\times\mathcal{Y}$$ with

1. $$\mathrm{Ob}(\mathcal{X}\times\mathcal{Y}):=\mathrm{Ob}(\mathcal{X})\mathrm{Ob}(\times\mathcal{Y})$$,

2. $$(\mathcal{X}\times\mathcal{Y})((x,y),(x',y')):=\mathcal{X}(x,x')\times\mathcal{Y}(y,y')$$, for two objects $$(x,y)$$ and $$(x',y')$$ in $$\mathrm{Ob}(\mathcal{X}\times\mathcal{Y})$$.

According to the definition (Def. 2.43) of a $$\mathcal{V}$$-category for all $$x,x'\in\mathrm{Ob}(\mathcal{X})$$, $$I\leq\mathcal{X}(x,x')$$.

1. $$(\mathcal{X}\times\mathcal{Y})((x,y),(x',y')):=\mathcal{X}(x,x')\otimes\mathcal{Y}(y,y')$$, by definition of a $$\mathcal{V}$$-product. $$I\otimes I\leq\mathcal{X}(x,x')\times\mathcal{Y}(y,y')$$, by definition of a $$\mathcal{V}$$-category and the definition of a product preorder (Ex. 1.47). Since $$\mathcal{V}$$ is a symmetric monoidal preorder, $$I\otimes I=I$$. This shows that $$I\leq(\mathcal{X}\times\mathcal{Y})((x,y),(x',y'))$$, for all $$(x,y),(x',y')\in\mathrm{Ob}(\mathcal{X}\times\mathcal{Y})$$, so also, in particular, for $$x',y')=(x,y)$$.

2. $$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))=\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)$$ and $$(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))=\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)$$, so $$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))\otimes(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))$$ $$=\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)$$. It is sufficient that $$\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)$$, to show that $$\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)$$ $$\leq\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{Y}(y_2,y_3)$$. By definition of a $$\mathcal{V}$$-category, $$\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_1,x_3)$$ and $$\mathcal{Y}(x_1,x_2)\otimes\mathcal{Y}(x_2,x_3)\leq\mathcal{Y}(x_1,x_3)$$, so $$\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{Y}(y_2,y_3)\leq\mathcal{X}(x_1,x_3)\otimes\mathcal{Y}(y_1,y_3)$$ $$=(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_3,y_3))$$. Which shows that $$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))\otimes(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))$$ $$\leq(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_3,y_3))$$.

3. The symmetry condition of Definition 2.2 d) is used in order to satisfy the requirement that $$\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)$$ (by requiring this condition with equality).

Comment Source:**Definition 2.71**: Let \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$ be \$$\mathcal{V}\$$-categories. Define their \$$\mathcal{V}\$$_-product_, or simply _product_, to be the \$$\mathcal{V}\$$-category \$$\mathcal{X}\times\mathcal{Y}\$$ with 1. \$$\mathrm{Ob}(\mathcal{X}\times\mathcal{Y}):=\mathrm{Ob}(\mathcal{X})\mathrm{Ob}(\times\mathcal{Y})\$$, 2. \$$(\mathcal{X}\times\mathcal{Y})((x,y),(x',y')):=\mathcal{X}(x,x')\times\mathcal{Y}(y,y')\$$, for two objects \$$(x,y)\$$ and \$$(x',y')\$$ in \$$\mathrm{Ob}(\mathcal{X}\times\mathcal{Y})\$$. According to the definition (Def. 2.43) of a \$$\mathcal{V}\$$-category for all \$$x,x'\in\mathrm{Ob}(\mathcal{X})\$$, \$$I\leq\mathcal{X}(x,x')\$$. 1. \$$(\mathcal{X}\times\mathcal{Y})((x,y),(x',y')):=\mathcal{X}(x,x')\otimes\mathcal{Y}(y,y')\$$, by definition of a \$$\mathcal{V}\$$-product. \$$I\otimes I\leq\mathcal{X}(x,x')\times\mathcal{Y}(y,y')\$$, by definition of a \$$\mathcal{V}\$$-category and the definition of a product preorder (Ex. 1.47). Since \$$\mathcal{V}\$$ is a symmetric monoidal preorder, \$$I\otimes I=I\$$. This shows that \$$I\leq(\mathcal{X}\times\mathcal{Y})((x,y),(x',y'))\$$, for all \$$(x,y),(x',y')\in\mathrm{Ob}(\mathcal{X}\times\mathcal{Y})\$$, so also, in particular, for \$$x',y')=(x,y)\$$. 2. \$$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))=\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)\$$ and \$$(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))=\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)\$$, so \$$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))\otimes(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))\$$ \$$=\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)\$$. It is sufficient that \$$\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\$$, to show that \$$\mathcal{X}(x_1,x_2)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_2,y_3)\$$ \$$\leq\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{Y}(y_2,y_3)\$$. By definition of a \$$\mathcal{V}\$$-category, \$$\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_1,x_3)\$$ and \$$\mathcal{Y}(x_1,x_2)\otimes\mathcal{Y}(x_2,x_3)\leq\mathcal{Y}(x_1,x_3)\$$, so \$$\mathcal{X}(x_1,x_2)\otimes\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\otimes\mathcal{Y}(y_2,y_3)\leq\mathcal{X}(x_1,x_3)\otimes\mathcal{Y}(y_1,y_3)\$$ \$$=(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_3,y_3))\$$. Which shows that \$$(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_2,y_2))\otimes(\mathcal{X}\times\mathcal{Y})((x_2,y_2),(x_3,y_3))\$$ \$$\leq(\mathcal{X}\times\mathcal{Y})((x_1,y_1),(x_3,y_3))\$$. 3. The symmetry condition of Definition 2.2 d) is used in order to satisfy the requirement that \$$\mathcal{Y}(y_1,y_2)\otimes\mathcal{X}(x_2,x_3)\leq\mathcal{X}(x_2,x_3)\otimes\mathcal{Y}(y_1,y_2)\$$ (by requiring this condition with equality).