Axiom:Axioms for Morphisms-Only Category Theory

Axiom

Let $\mathcal {MOCT}$ be the language of (morphisms-only) category theory.

Then (morphisms-only) category theory is the mathematical theory arising from the following axioms:

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\((\text {MOCT} 0)\)	$:$			\(\ds \forall x, y, z, z':\)	\(\ds \paren {\map {R_\circ} {x, y, z} \land \map {R_\circ} {x, y, z'} } \implies z = z' \)				$\circ$ 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 $\circ$

Also see

• Definition:Morphisms-Only Metacategory, a metamodel for these axioms