Diagram (category Theory) - Examples

Examples

• Given any object A in C, one has the constant diagram, which is the diagram that maps all objects in J to A, and all morphisms of J to the identity morphism on A. Notationally, one often uses an underbar to denote the constant diagram: thus, for any object in C, one has the constant diagram .

• If J is a (small) discrete category, then a diagram of type J is essentially just an indexed family of objects in C (indexed by J). When used in the construction of the limit, the result is the product; for the colimit, one gets the coproduct. So, for example, when J is the discrete category with two objects, the resulting limit is just the binary product.

• If J = -1 ← 0 → +1, then a diagram of type J (A ← B → C) is a span, and its colimit is a pushout. If one were to "forget" that the diagram had object B and the two arrows B → A, B → C, the resulting diagram would simply be the discrete category with the two objects A and C, and the colimit would simply be the binary coproduct. Thus, this example shows an important way in which the idea of the diagram generalizes that of the index set in set theory: by including the morphisms B → A, B → C, one discovers additional structure in constructions built from the diagram, structure that would not be evident if one only had an index set with no relations between the objects in the index.

• If J = -1 → 0 ← +1, then a diagram of type J (A → B ← C) is a cospan, and its limit is a pullback.

• The index is called "two parallel morphisms", or sometimes the free quiver or the walking quiver. A diagram of type J is then a quiver; its limit is an equalizer, and its colimit is a coequalizer.

• If J is a poset category, then a diagram of type J is a family of objects D_i together with a unique morphism f_ij : D_i → D_j whenever i ≤ j. If J is directed then a diagram of type J is called a direct system of objects and morphisms. If the diagram is contravariant then it is called an inverse system.