Definition:Diagram (Category Theory)

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.