Editing Commutative diagram (section)

== Diagrams as functors ==
{{Main|Diagram (category theory)}}

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 (category theory)|diagram]].'''

More formally, a commutative diagram is a visualization of a diagram indexed by a [[Posetal category|poset category]]. Such a diagram typically includes:
* 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),
* the commutativity of the diagram (the equality of different compositions of maps between two objects), corresponding to the uniqueness of a map between two objects in a poset category.

Conversely, given a commutative diagram, it defines a poset category, where:
* 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). As a simple example, the diagram of a single object with an endomorphism (<math>f\colon X \to X</math>), or with two parallel arrows (<math>\bullet \rightrightarrows \bullet</math>, that is, <math>f,g\colon X \to Y</math>, sometimes called the [[free quiver]]), as used in the definition of [[equaliser (mathematics)|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).