# Category Axioms are Self-Dual

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## Contents

## 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

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 3.1$