Characterization of Metacategory via Equations

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Theorem

Let $\mathbf C_0$ and $\mathbf C_1$ be collections of objects.

Let $\operatorname{cod}$ and $\operatorname{dom}$ assign to every element of $\mathbf C_1$ an element of $\mathbf C_0$.

Let $\operatorname{id}$ assign to every element of $\mathbf C_0$ an element of $\mathbf C_1$.


Denote with $\mathbf C_2$ the collection of pairs $\left({f, g}\right)$ of elements of $\mathbf C_1$ satisfying:

$\operatorname{dom} g = \operatorname{cod} f$

Let $\circ$ assign to every such pair an element of $\mathbf C_1$.


Then $\mathbf C_0, \mathbf C_1, \operatorname{cod}, \operatorname{dom}, \operatorname{id}$ and $\circ$ together determine a metacategory $\mathbf C$ iff the following seven axioms are satisfied:

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

with $A$ and $f,g,h$ arbitrary elements of $\mathbf C_0$ and $\mathbf C_1$, respectively.

Further, in the last two lines it is presumed that all compositions are defined.


Hence it follows that:


Proof


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