Category Axioms are Self-Dual

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Theorem

Morphisms-Only Category Theory

Let $\mathrm{MOCT}$ be the collection of axioms for morphisms-only category theory.


Then:

$\mathrm{MOCT} = \mathrm{MOCT}^*$

where $\mathrm{MOCT}^*$ consists of the dual statements of those in $\mathrm{MOCT}$.


Object Category Theory

Let $\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations.


Then:

$\mathrm{CT} = \mathrm{CT}^*$

where $\mathrm{CT}^*$ consists of the dual statements of those in $\mathrm{CT}$.


Proof

Proof for Morphisms-Only Category Theory


Proof for Object Category Theory

The seven axioms are:

\(\displaystyle \operatorname{dom} \operatorname{id}_A = A\) \(\qquad\) \(\displaystyle \operatorname{cod} \operatorname{id}_A = A\)
\(\displaystyle f \circ 1_{\operatorname{dom} f} = f\) \(\) \(\displaystyle 1_{\operatorname{cod} f} \circ f = f\)
\(\displaystyle \operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f\) \(\) \(\displaystyle \operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g\)
\(\displaystyle h \circ \left({g \circ f}\right)\) \(=\) \(\displaystyle \left({h \circ g}\right) \circ f\)

Their duals are:

\(\displaystyle \operatorname{cod} \operatorname{id}_A = A\) \(\qquad\) \(\displaystyle \operatorname{dom} \operatorname{id}_A = A\)
\(\displaystyle 1_{\operatorname{cod} f} \circ f = f\) \(\) \(\displaystyle f \circ 1_{\operatorname{dom} f} = f\)
\(\displaystyle \operatorname{cod} \left({f \circ g}\right) = \operatorname{cod} f\) \(\) \(\displaystyle \operatorname{dom} \left({f \circ g}\right) = \operatorname{dom} g\)
\(\displaystyle \left({f \circ g}\right) \circ h\) \(=\) \(\displaystyle f \circ \left({g \circ h}\right)\)

It is seen that only names of the bound variables $f,g$ and $h$ have been changed at some places.

Therefore, we conclude:

$\mathrm{CT}^* = \mathrm{CT}$

$\blacksquare$


Sources