Definition:Morphisms-Only Metacategory

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A morphisms-only metacategory is a metamodel for the language of category theory subject to the following axioms:

\((\text {MOCT} 0)\)   $:$     \(\ds \forall x, y, z, z':\) \(\ds \paren {\map {R_\circ} {x, y, z} \land \map {R_\circ} {x, y, z'} } \implies z = z' \)      $\circ$ is a partial mapping in two variables
\((\text {MOCT} 1)\)   $:$     \(\ds \forall x, y:\) \(\ds \Dom x = \Cdm y \iff \exists z: \map {R_\circ} {x, y, z} \)      domain of composition $\circ$
\((\text {MOCT} 2)\)   $:$     \(\ds \forall x, y, z:\) \(\ds \map {R_\circ} {x, y, z} \implies \paren {\Dom z = \Dom y \land \Cdm z = \Cdm x} \)      Domain and codomain of a composite $z = x \circ y$
\((\text {MOCT} 3)\)   $:$     \(\ds \forall x, y, z, a, b:\) \(\ds \map {R_\circ} {x, y, a} \land \map {R_\circ} {y, z, b} \implies \paren {\exists w: \map {R_\circ} {x, b, w} \land \map {R_\circ} {a, z, w} } \)      $\circ$ is associative
\((\text {MOCT} 4)\)   $:$     \(\ds \forall x:\) \(\ds \map {R_\circ} {x, \Dom x, x} \land \map {R_\circ} {\Cdm x, x, x} \)      Left identity and right identity for $\circ$


A morphisms-only metacategory can thus be described as follows.

Let $\mathbf C_1$ be a collection of objects called morphisms.

Let $\mathbf C_2$ be the collection of pairs of morphisms $\left({g, f}\right)$ with $\operatorname{cod} f = \operatorname{dom} g$; write $\mathbf C_2 \left({g, f}\right)$ to express that $\left({g, f}\right)$ is a member of $\mathbf C_2$.

By $(MOCT1)$, we see that $\mathbf C_2 \left({g, f}\right)$ thus is an abbreviation of the statement $\exists h: R_\circ \left({g, f, h}\right)$.

Let $\circ$ be an operation symbol which must assign to every pair of morphisms $\left({g, f}\right)$ in $\mathbf C_2$ a morphism $g \circ f$, called the composition of $g$ with $f$.

We see that $g \circ f$ satisfies $R_\circ \left({g, f, g \circ f}\right)$; by axiom $(MOCT0)$, it is unique.

Axioms $(MOCT1)$ up to $(MOCT3)$ combine to ensure that $h \circ \left({g \circ f}\right)$ is defined iff $\left({h \circ g}\right) \circ f$ is, and that they are equal when this is the case.

Finally, axiom $(MOCT4)$ entails the existence and uniqueness of left- and right-identities for $\circ$.

Also see