A [[Category]] $\mathcal C$ is some collection of objects with [[Function|arrows]] between objects that compose [[Associative|associatively]]. >[!note] See Also: [[Category Theory]] ### Formal The more formal definition is that a [[Category]] $\mathcal C$ must contain: - A [[./Class|class]] of objects $\op{ob}(\mathcal C)$ - A [[Class|class]] of [[Morphism|Morphisms]] (arrows) $\op{mor(\mathcal C)}$. - A form of [[Associative]] [[Composition]] between [[Morphism|Morphisms]] Given [[Universal Quantifier|any]] two [[Morphism|Morphisms]] $f,g\in\op{mor}(\mathcal C)$ and [[Universal Quantifier|any]] pair of three objects $x,y,z\in\op{ob}(\mathcal C)$, they must satisfy: $\huge \begin{align} f: x &\mapsto y \\ g: y &\mapsto z \\ f \circ g &: x \mapsto z \end{align} $ ![[../../00 Asset Bank/Pasted image 20250419002923.png|invert_S]]