# Definition:Commutative Diagram

## Definition

A **commutative diagram** is a graphical technique designed to illustrate the construction of composite mappings.

It is also a widespread tool in category theory, where it deals with morphisms instead of mappings.

- $\begin{xy} \xymatrix@L+2mu@+1em{ S_1 \ar[r]^*{f_1} \ar@{-->}[rd]_*{f_2 \circ f_1} & S_2 \ar[d]^*{f_2} \\ & S_3 }\end{xy}$

It consists of:

- $(1): \quad$ A collection of points representing the various domains and codomains of the mappings in question
- $(2): \quad$ Arrows representing the mappings themselves.

The diagram is properly referred to as **commutative** if and only if all the various paths from the base of one arrow to the head of another represent equal mappings.

A mapping which is uniquely determined by the rest of the diagram may be indicated by a dotted arrow.

It is however generally advisable not to use more than one dotted arrow per diagram, so as to avoid confusion.

### Relations

A similar technique can be applied to composition of relations in general, but this is rarely seen.

## Also known as

Some sources refer to a diagram of this nature as a **function diagram**.

Some refer to it as just a **diagram**.

## Examples

### Arbitrary Example

Let $A, B, X, Y$ be arbitrary sets.

Let:

\(\ds f: \, \) | \(\ds A\) | \(\to\) | \(\ds X\) | |||||||||||

\(\ds g: \, \) | \(\ds B\) | \(\to\) | \(\ds Y\) | |||||||||||

\(\ds \alpha: \, \) | \(\ds A\) | \(\to\) | \(\ds B\) | |||||||||||

\(\ds \beta: \, \) | \(\ds X\) | \(\to\) | \(\ds Y\) |

be mappings such that:

- $\beta \circ f = g \circ \alpha = k$

where $\circ$ denotes composition of mappings.

This can be depicted using the following commutative diagram:

- $\begin{xy} \xymatrix@L+2mu@+1em{ A \ar[r]^*{\alpha} \ar[d]_*{f} \ar[rd]^*{k} & B \ar[d]^*{g} \\ X \ar[r]^*{\beta} & Y }\end{xy}$

### Square Function with Square Root

Let $g$ and $h$ be the real functions defined as:

- $\forall x \in \R: \map g x = x^2$

- $\forall x \in \R_{\ge 0}: \map h x = \sqrt x$

The composition $h \circ g$ can be depicted using a commutative diagram as follows:

- $\begin{xy} \xymatrix@L+2mu@+1em{ \R \ar[r]^*{g} \ar@{-->}[rd]_*{h \circ g} & \R_{\ge 0} \ar[d]^*{h} \\ & \R_{\ge 0} }\end{xy}$

## Also see

- Results about
**commutative diagrams**can be found**here**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 3.4$. Product of mappings: Figure $11$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.14$: Composition of Functions: Theorem $14.5$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Composition of functions - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 8$: Composition of Functions and Diagrams - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**commutative diagram** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**commutative diagram**