Characterization of Metacategory via Equations

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Theorem

Let $\mathbf C_0$ and $\mathbf C_1$ be collections of objects.

Let $\operatorname{Cdm}$ and $\operatorname{Dom}$ assign to every element of $\mathbf C_1$ an element of $\mathbf C_0$.

Let $\operatorname{id}$ assign to every element of $\mathbf C_0$ an element of $\mathbf C_1$.

Denote with $\mathbf C_2$ the collection of pairs $\tuple {f, g}$ of elements of $\mathbf C_1$ satisfying:

$\Dom g = \Cdm f$

Let $\circ$ assign to every such pair an element of $\mathbf C_1$.

Then $\mathbf C_0, \mathbf C_1, \operatorname{Cdm}, \operatorname{Dom}, \operatorname{id}$ and $\circ$ together determine a metacategory $\mathbf C$ if and only if the following seven axioms are satisfied:

\(\ds \Dom {\operatorname{id}_A} = A\)

\(\qquad\)

\(\ds \Cdm {\operatorname{id}_A} = A\)

\(\ds f \circ \operatorname{id}_{\Dom f} = f\)

\(\)

\(\ds \operatorname{id}_{\Cdm f} \circ f = f\)

\(\ds \Dom {g \circ f} = \Dom f\)

\(\)

\(\ds \Cdm {g \circ f} = \Cdm g\)

\(\ds h \circ \paren {g \circ f}\)

\(=\)

\(\ds \paren {h \circ g} \circ f\)

where $A$ and $f, g, h$ are arbitrary elements of $\mathbf C_0$ and $\mathbf C_1$, respectively.

Further, in the last two lines, it is presumed that all compositions are defined.

Hence it follows that:

• $\mathbf C_0$ and $\mathbf C_1$ represent the collections of objects and morphisms of $\mathbf C$
• $\operatorname{Dom}$ and $\operatorname{Cdm}$ represent the domain and codomain of a morphism of $\mathbf C$
• $\operatorname{id}$ represents the identity morphisms of $\mathbf C$
• $\mathbf C_2$ represents the collection of composable morphisms of $\mathbf C$
• $\circ$ represents the composition of morphisms in $\mathbf C$

Proof

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Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (next): $\S 3.1$