Diagram (category Theory) - Commutative Diagrams

Commutative Diagrams

Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.

Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism, or with two parallel arrows (; ) need not commute. Further, diagrams may be impossible to draw (because infinite) or simply messy (because too many objects or morphisms); however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.