Axiom:Axioms for Morphisms-Only Category Theory

From ProofWiki
Jump to navigation Jump to search


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:

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