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
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Proof for Object Category Theory
The seven axioms are:
| \(\displaystyle \operatorname{dom} \operatorname{id}_A = A\) | \(\qquad\) | \(\displaystyle \operatorname{cod} \operatorname{id}_A = A\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle f \circ 1_{\operatorname{dom} f} = f\) | \(\) | \(\displaystyle 1_{\operatorname{cod} f} \circ f = f\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f\) | \(\) | \(\displaystyle \operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle h \circ \left({g \circ f}\right)\) | \(=\) | \(\displaystyle \left({h \circ g}\right) \circ f\) | $\quad$ | $\quad$ |
Their duals are:
| \(\displaystyle \operatorname{cod} \operatorname{id}_A = A\) | \(\qquad\) | \(\displaystyle \operatorname{dom} \operatorname{id}_A = A\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle 1_{\operatorname{cod} f} \circ f = f\) | \(\) | \(\displaystyle f \circ 1_{\operatorname{dom} f} = f\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \operatorname{cod} \left({f \circ g}\right) = \operatorname{cod} f\) | \(\) | \(\displaystyle \operatorname{dom} \left({f \circ g}\right) = \operatorname{dom} g\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \left({f \circ g}\right) \circ h\) | \(=\) | \(\displaystyle f \circ \left({g \circ h}\right)\) | $\quad$ | $\quad$ |
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
- 2010: Steve Awodey: Category Theory ... (previous) ... (next): $\S 3.1$
