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:

\((MOCT0)\)   $:$     \(\ds \forall x,y,z,z':\) \(\ds \left({R_\circ \left({x, y, z}\right) \land R_\circ \left({x, y, z'}\right)}\right) \implies z = z' \)             $\circ$ is a partial mapping in two variables
\((MOCT1)\)   $:$     \(\ds \forall x,y:\) \(\ds \operatorname{dom} x = \operatorname{cod} y \iff \exists z: R_\circ \left({x, y, z}\right) \)             domain of composition $\circ$
\((MOCT2)\)   $:$     \(\ds \forall x,y,z:\) \(\ds R_\circ \left({x, y, z}\right) \implies \left({\operatorname{dom} z = \operatorname{dom} y \land \operatorname{cod} z = \operatorname{cod} x}\right) \)             Domain and codomain of a composite $z = x \circ y$
\((MOCT3)\)   $:$     \(\ds \forall x,y,z,a,b:\) \(\ds R_\circ \left({x, y, a}\right) \land R_\circ \left({y, z, b}\right) \implies \left({\exists w: R_\circ \left({x, b, w}\right) \land R_\circ \left({a, z, w}\right)}\right) \)             $\circ$ is associative
\((MOCT4)\)   $:$     \(\ds \forall x:\) \(\ds R_\circ \left({x, \operatorname{dom} x, x}\right) \land R_\circ \left({\operatorname{cod} x, x, x}\right) \)             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