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:
\((MOCT0)\) | $:$ | \(\displaystyle \forall x,y,z,z':\) | \(\displaystyle \left({R_\circ \left({x, y, z}\right) \land R_\circ \left({x, y, z'}\right)}\right) \implies z = z' \) | $\circ$ is a partial mapping in two variables | ||||
\((MOCT1)\) | $:$ | \(\displaystyle \forall x,y:\) | \(\displaystyle \operatorname{dom} x = \operatorname{cod} y \iff \exists z: R_\circ \left({x, y, z}\right) \) | domain of composition $\circ$ | ||||
\((MOCT2)\) | $:$ | \(\displaystyle \forall x,y,z:\) | \(\displaystyle R_\circ \left({x, y, z}\right) \implies \left({\operatorname{dom} z = \operatorname{dom} y \land \operatorname{cod} z = \operatorname{cod} x}\right) \) | Domain and codomain of a composite $z = x \circ y$ | ||||
\((MOCT3)\) | $:$ | \(\displaystyle \forall x,y,z,a,b:\) | \(\displaystyle R_\circ \left({x, y, a}\right) \land R_\circ \left({y, z, b}\right) \implies \left({\exists w: R_\circ \left({x, b, w}\right) \land R_\circ \left({a, z, w}\right)}\right) \) | $\circ$ is associative | ||||
\((MOCT4)\) | $:$ | \(\displaystyle \forall x:\) | \(\displaystyle R_\circ \left({x, \operatorname{dom} x, x}\right) \land R_\circ \left({\operatorname{cod} x, x, x}\right) \) | Left-identity and right-identity for $\circ$ |
Also see
- Morphisms-Only Metacategory, a metamodel for these axioms