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Exercise 72 - Chapter 2

edited June 2018 in Exercises

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).
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