Commutative diagram

Examples

Example 1

In the left diagram, which expresses the first isomorphism theorem, commutativity of the triangle means that ${\displaystyle f=\tilde{f}\circ \pi }$. In the right diagram, commutativity of the square means ${\displaystyle h\circ f=k\circ g}$.

Example 2

In order for the diagram below to commute, three equalities must be satisfied:

1. ${\displaystyle r\circ h\circ g=H\circ G\circ l}$
2. ${\displaystyle m\circ g=G\circ l}$
3. ${\displaystyle r\circ h=H\circ m}$

Here, since the first equality follows from the last two, it suffices to show that (2) and (3) are true in order for the diagram to commute. However, since equality (3) generally does not follow from the other two, it is generally not enough to have only equalities (1) and (2) if one were to show that the diagram commutes.

Diagram chasing

Diagram chasing (also called diagrammatic search) is a method of mathematical proof used especially in homological algebra, where one establishes a property of some morphism by tracing the elements of a commutative diagram. A proof by diagram chasing typically involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.

Examples of proofs by diagram chasing include those typically given for the five lemma, the snake lemma, the zig-zag lemma, and the nine lemma.

In higher category theory

In higher category theory, one considers not only objects and arrows, but arrows between the arrows, arrows between arrows between arrows, and so on ad infinitum. For example, the category of small categories Cat is naturally a 2-category, with functors as its arrows and natural transformations as the arrows between functors. In this setting, commutative diagrams may include these higher arrows as well, which are often depicted in the following style: ${\displaystyle \Rightarrow }$. For example, the following (somewhat trivial) diagram depicts two categories C and D, together with two functors F, G : C → D and a natural transformation α : F ⇒ G:

There are two kinds of composition in a 2-category (called vertical composition and horizontal composition), and they may also be depicted via pasting diagrams (see 2-category#Definitions for examples).

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. 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 (${\displaystyle f:X\to X}$), or with two parallel arrows (${\displaystyle \bullet \rightrightarrows \bullet }$, that is, ${\displaystyle f,g:X\to Y}$, 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).