Definition:Composition of Mappings/Commutative Diagram

Definition

Let $S_1$, $S_2$ and $S_3$ be sets.

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.

The concept of composition of mappings can be illustrated by means of a commutative diagram.

This diagram illustrates the specific example of $f_2 \circ f_1$:

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

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.4$. Product of mappings: Figure $11$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Composition of functions
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 24$: Composition of Mappings
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.2$

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• 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions