nLab
diagram

Definition

A diagram in a category $C$ is simply a functor $F:J\to C$. The category $J$ is called the shape or index category of the diagram, and is typically understood to be a small category.

For example:

• Every category $C$ admits a unique diagram whose shape is the empty (initial) category, which is called the empty diagram in $C$.
• Specifying a diagram in $C$ whose shape is the terminal category $1$ is the same as specifying an object of $C$, the image of the unique object of $1$. (See global element)
• A morphism in $C$ is a diagram whose shape is a category with two objects and a single morphism from one to the other.
• A pair of objects is a diagram whose shape is a discrete category with two objects.
• A pair of parallel morphisms is a diagram whose shape is a category with two objects and two morphisms from one to the other.
• A span is a diagram whose shape is a category with just three objects and single morphisms from one of the objects to the other two; dually, a cospan is a diagram whose shape is opposite to the shape of a span.

See also limit, colimit.