Definition:Metacategory

Definition
A metacategory is a metagraph subject to extra restrictions.

In particular a metacategory $\mathcal C$ consists of:
 * objects $X, Y, Z, \ldots$
 * morphisms (or arrows or maps) $f, g, h, \ldots$ subject to:
 * 1. Composition: for objects $X,Y,Z$ and morphisms $X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z$, there exists a morphism:


 * $g \circ f : X \to Z$


 * called the composition of $f$ and $g$.


 * 2. Identity: for every object $X$ there is a morphism $\operatorname{id}_X : X \to X$ such that for any object $Y$, and any morphisms $f : X \to Y$, $g : Y \to X$:


 * $f \circ \operatorname{id}_X = f$, and $\operatorname{id}_X \circ g = g$


 * called the identity morphism.


 * 3. Associativity: For any three morphisms $f,g,h$:


 * $f \circ (g \circ h) = (f \circ g) \circ h$


 * whenever the compositions are defined (as determined by 1.).

A metacategory is purely axiomatic, and does not use set theory.

For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory.