# Definition:Diagram (Category Theory)

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## Definition

Let $\mathbf J$ and $\mathbf C$ be metacategories.

A **diagram of type $\mathbf J$** in $\mathbf C$ is a functor $D: \mathbf J \to \mathbf C$.

### Index Category

In this context, $\mathbf J$ is referred to as the **index category**.

Its objects are typically denoted by lowercase letters, $i, j$ etc.

Furthermore, one writes $D_i$ in place of the formally more correct $D \left({i}\right)$.

Similarly, for $\alpha: i \to j$ a morphism one writes $D_\alpha$ in place of $D \left({\alpha}\right)$.

## Also known as

It is sometimes more convenient to refer to a **diagram of type $\mathbf J$** as a **$\mathbf J$-diagram**.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 5.4$: Definition $5.15$