Duality Principle (Category Theory)/Conceptual Duality

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This proof is about Duality Principle in the context of Category Theory. For other uses, see Duality Principle.

Theorem

Let $\Sigma$ be a statement about metacategories, be it in natural language or otherwise.

Suppose that $\Sigma$ holds for all metacategories.


Then so does its dual statement $\Sigma^*$.


Proof

From Dual Category of Dual Category, any metacategory $\mathbf C$ may be regarded as a dual category.

Thus it is sufficient to verify $\Sigma^*$ holds in all dual categories.


In any dual category $\mathbf C^{\text{op}}$, we see that:

$\operatorname{dom} f^{\text{op}} = \left({\operatorname{cod} f}\right)^{\text{op}}$
$\operatorname{cod} f^{\text{op}} = \left({\operatorname{dom} f}\right)^{\text{op}}$
$f^\text{op} \circ g^\text{op} = \left({g \circ f}\right)^{\text{op}}$

Thus, the interpretation of $\Sigma^*$ in $\mathbf C^{\text{op}}$ comes down to the interpretation of $\Sigma$ in $\mathbf C$.

That is, except for amending every symbol $\star$ of $\mathbf C$ with a superscript "$\text{op}$", yielding $\star^\text{op}$.


This operation does not change the validity of $\Sigma$ in any possible way, and $\Sigma$ was assumed to hold in $\mathbf C$.

Hence we conclude that $\Sigma^*$ must hold in $\mathbf C^{\text{op}}$ as well.


The result follows.

$\blacksquare$


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