Axiom:Axioms for Morphisms-Only Category Theory
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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