# Definition:Metacategory

## Definition

A metacategory is a metagraph subject to extra restrictions.

As such, a metacategory $\mathbf C$ consists of:

• A collection $\mathbf C_0$ of objects $X, Y, Z, \ldots$
• A collection $\mathbf C_1$ of morphisms $f, g, h, \ldots$ between its objects

The morphisms of $\mathbf C$ are subjected to:

 $(C1)$ $:$ Composition For objects $X, Y, Z$ and morphisms $X \stackrel {f} {\longrightarrow} Y \stackrel {g} {\longrightarrow} Z$ with $\operatorname{cod} f = \operatorname{dom} \, g$, there exists a morphism: $g \circ f : X \to Z$ called the composition of $f$ and $g$. $(C2)$ $:$ Identity For every object $X$, there is a morphism ${\operatorname{id}_X}: X \to X$, called the identity morphism, such that: $f \circ {\operatorname{id}_X} = f$, and ${\operatorname{id}_X} \circ g = g$ for any object $Y$ and morphisms $f : X \to Y$ and $g : Y \to X$ $(C3)$ $:$ Associativity For any three morphisms $f, g, h$: $f \circ \left({g \circ h}\right) = \left({f \circ g}\right) \circ h$ whenever these compositions are defined (according to $(C1)$).

To describe a metacategory it is necessary to specify:

However, the last two are often taken to be implicit when the objects and morphisms are familiar.

Of course, after defining these, it is still to be shown that $(C1)$ up to $(C3)$ are satisfied.

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.

For some purposes, it is convenient to have a different description of a metacategory. Two such descriptions are found on:

## Also denoted as

Many conventions exist to denote metacategories.

For example, many authors prefer a calligraphic $\mathcal C$, scripted $\mathscr C$ or a fraktur $\mathfrak C$ over the simple bold $\mathbf C$.

Alternatively, the collection of objects $\mathbf C_0$ can be denoted $\operatorname{ob} \mathbf C$.

The collection of morphisms $\mathbf C_1$ is also denoted $\operatorname{mor} \mathbf C$, $\operatorname{Hom}_{\mathbf C}$ and $\operatorname{Hom} \mathbf C$.

Also, although formally undefined, it is often convenient to write $X \in \mathbf C_0$ in place of '$X$ is an object of $\mathbf C$'.

Similarly, the notation $f \in \mathbf C_1$ may be encountered.