Commutative Diagram - Diagrams As Functors

Diagrams As Functors

A commutative diagram in a category C can be interpreted as a functor from an index category J to C; one calls the functor a diagram.

More formally, a commutative diagram is a visualization of a diagram indexed by a poset category:

• one draws a node for every object in the index category,
• an arrow for a generating set of morphisms,

  omitting identity maps and morphisms that can be expressed as compositions,

• and the commutativity of the diagram (the equality of different compositions of maps between two objects) corresponds to the uniqueness of a map between two objects in a poset category.

Conversely, given a commutative diagram, it defines a poset category:

• the objects are the nodes,
• there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
• with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).

However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram): most simply, the diagram of a single object with an endomorphism, or with two parallel arrows (, that is, sometimes called the free quiver), as used in the definition of equalizer need not commute. Further, diagrams may be messy or impossible to draw when the number of objects or morphisms is large (or even infinite).