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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.

$$\newcommand{\cat}{\mathcal{#1}}\def\Ob{{\mathrm{Ob}}}\Ob(P)$$ 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}{\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).