Category Axioms are Self-Dual/Object Category Theory

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Theorem

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

The seven axioms are:

\(\ds \operatorname {dom} \operatorname {id}_A = A\) \(\qquad\) \(\ds \operatorname {cod} \operatorname {id}_A = A\)
\(\ds f \circ 1_{\operatorname {dom} f} = f\) \(\) \(\ds 1_{\operatorname {cod} f} \circ f = f\)
\(\ds \map {\operatorname {dom} } {g \circ f} = \operatorname{dom} f\) \(\) \(\ds \map {\operatorname {cod} } {g \circ f} = \operatorname{cod} g\)
\(\ds h \circ \paren {g \circ f}\) \(=\) \(\ds \paren {h \circ g} \circ f\)

Their duals are:

\(\ds \operatorname {cod} \operatorname {id}_A = A\) \(\qquad\) \(\ds \operatorname {dom} \operatorname {id}_A = A\)
\(\ds 1_{\operatorname {cod} f} \circ f = f\) \(\) \(\ds f \circ 1_{\operatorname {dom} f} = f\)
\(\ds \map {\operatorname {cod} } {f \circ g} = \operatorname {cod} f\) \(\) \(\ds \map {\operatorname {dom} } {f \circ g} = \operatorname {dom} g\)
\(\ds \paren {f \circ g} \circ h\) \(=\) \(\ds f \circ \paren {g \circ h}\)

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


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