Definition:Morphisms-Only Metacategory

Definition

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 $\circ$

Explanation

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 $\circ$.