Axiom:Axioms for Morphisms-Only Category Theory
Jump to navigation
Jump to search
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:
This article needs to be linked to other articles. In particular: Mathematical theory here refers to something that we have not defined yet You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
\((\text {MOCT} 0)\) | $:$ | \(\ds \forall x, y, z, z':\) | \(\ds \paren {\map {R_\circ} {x, y, z} \land \map {R_\circ} {x, y, z'} } \implies z = z' \) | $\circ$ 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 $\circ$ |
This page has been identified as a candidate for refactoring of basic complexity. In particular: Separate the m parameters of the above into ml, no, mr so that the operators line up neatly, or as appropriate Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
Also see
- Definition:Morphisms-Only Metacategory, a metamodel for these axioms
Sources
There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page. |