Definition:Composition of Functors

Definition

Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be (covariant) functors.

The composition of $G$ with $F$ is the functor $GF: \mathbf C \to \mathbf E$ defined by:

			For all objects $C$ of $\mathbf C$:			\(\ds \map {GF} C \)	\(\ds := \)	\(\ds \map G {FC} \)
			For all morphisms $f: C_1 \to C_2$ of $\mathbf C$:			\(\ds \map {GF} f \)	\(\ds := \)	\(\ds \map G {Ff} \)

$GF$ is said to be a composite functor.

Presentational Note

Because of the way $GF$ is defined, when a working knowledge of functors is assumed, the brackets used in defining $GF$ may be (and usually are) disposed of whenever possible.

This is justified by the definition above, and the result Composition of Functors is Associative.

In such situations, they can only hamper one's understanding of the expression.

Also known as

Some sources prefer to write $G \circ F$ in place of $GF$, explicitly putting the $\circ$ for composition.

Since expressions like:

$\map {\paren {G \circ F} } {g \circ f}$

are bound to lead to confusion, the $\circ$ for composition of functors is to be suppressed on ProofWiki.

Also see

• Composite Functor is Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.6$