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 $5.15$