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Exercise 46 - Chapter 4

\(\def\cat#1{{\mathcal{#1}}}\) Check that monoidal categories generalize monoidal preorders: a monoidal preorder is a monoidal category \((\cat{P},I,\otimes)\) where, for every \(p,q\in\cat{P}\), the set \(\cat{P}(p,q)\) has at most one element.

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  • 1.

    \(\newcommand{\cat}[1]{\mathcal{#1}}\def\Ob{{\mathrm{Ob}}}\Ob(P)\) serves as the set on which the preorder is defined. The functor \(\otimes:\cat{P}\times\cat{P}\to\cat{P}\) induces a function \(\otimes_\Ob:\Ob(\cat{P})\times\Ob(\cat{P})\to\Ob(\cat{P})\). Properties a) and b) in the definition of a monoidal category imply that property b) in the definition of a weak symmetric monoidal preorder holds. The existence of an associator natural isomorphism implies associativity in the weak monoidal preorder. The existence of a swap natural isomorphism guarantees symmetry holds in a weak monoidal preorder. The final property to check is monotonicity. Monotonicity in the weak monoidal preorder follows from the function on Hom-sets induced by \(\otimes\). In particular, if \(f\in\cat{P}(p,q)\) and \(g\in\cat{P}(x,y)\neq\varnothing\), then \(f\otimes_\mathrm{Mor}g\in\cat{P}(p\otimes_\Ob x,q\otimes_\Ob y)\) (in the case of a monoidal preorder, the fact that the product Hom-set is non-empty is exactly what is meant by an inequality holding between the corresponding two elements of the preorder).

    Comment Source:\\(\newcommand{\cat}[1]{\mathcal{#1}}\def\Ob{{\mathrm{Ob}}}\Ob(P)\\) serves as the set on which the preorder is defined. The functor \\(\otimes:\cat{P}\times\cat{P}\to\cat{P}\\) induces a function \\(\otimes\_\Ob:\Ob(\cat{P})\times\Ob(\cat{P})\to\Ob(\cat{P})\\). Properties a) and b) in the definition of a monoidal category imply that property b) in the definition of a weak symmetric monoidal preorder holds. The existence of an associator natural isomorphism implies associativity in the weak monoidal preorder. The existence of a swap natural isomorphism guarantees symmetry holds in a weak monoidal preorder. The final property to check is monotonicity. Monotonicity in the weak monoidal preorder follows from the function on Hom-sets induced by \\(\otimes\\). In particular, if \\(f\in\cat{P}(p,q)\\) and \\(g\in\cat{P}(x,y)\neq\varnothing\\), then \\(f\otimes_\mathrm{Mor}g\in\cat{P}(p\otimes_\Ob x,q\otimes\_\Ob y)\\) (in the case of a monoidal preorder, the fact that the product Hom-set is non-empty is exactly what is meant by an inequality holding between the corresponding two elements of the preorder).
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