Composition of Morphisms

In category theory, the composition of morphisms is a fundamental operation that combines two sequential morphisms, $ f $ and $ g $, to form a new morphism. $$ g \circ f $$

Imagine a category consisting of three objects $ A $, $ B $, and $ C $, and two morphisms, $ f $ and $ g $.

The morphism $ f $ maps from $ A $ to $ B $.

$$ f: \ A \rightarrow B $$

Meanwhile, the morphism $ g $ maps from $ B $ to $ C $.

$$ g: \ B \rightarrow C $$

Given that the codomain of $ f $ coincides with the domain of $ g $, we can combine these two morphisms into a single morphism, denoted $ g \circ f $.

At times, the composition of morphisms is simply denoted as $ gf $ when clarity allows.

This composite, $ g \circ f $, represents a new morphism from $ A $ to $ C $, achieved by applying $ f $ followed by $ g $.

$$ g \circ f: \ A \rightarrow C $$

In essence, in the composition $ g \circ f $, the function $ f $ is executed first, and its output then serves as the input for $ g $.

Properties of Morphism Composition

The composition of morphisms follows two key principles:

Associativity: If three morphisms exist, $ f: A \rightarrow B $, $ g: B \rightarrow C $, and $ h: C \rightarrow D $, then their composition adheres to the associative property: $ (h \circ g) \circ f = h \circ (g \circ f) $. This indicates that the outcome is consistent, regardless of how the operations are grouped.
Identity element: Every object $ A $ has an associated identity morphism $ 1_A: A \rightarrow A $ that acts as a neutral element in composition. Thus, any morphism $ f: A \rightarrow B $ composed with its identity yields $ f \circ 1_A = f $ and $ 1_B \circ f = f $.

The composition process can be visually depicted with a diagram:

$$ A \rightarrow B \rightarrow C $$

This diagram, where the direct arrow flows from $ A $ to $ C $ via $ B $, represents $ g \circ f $.

Example

Consider the 'Set' category, where objects are sets and morphisms are functions between these sets.

example of morphism composition

This category features three objects $ Obj(Set) = \{ A,B,D \} $ and two directed morphisms $ mor(Set) = \{f,g \} $.

The morphism $ f: A \rightarrow B $ is specified as:

$$ f(1) = a $$

$$ f(2) = a $$

$$ f(3) = b $$

The subsequent morphism $ g: B \rightarrow C $ is defined:

$$ g(a) = x $$

$$ g(b) = y $$

With the alignment of $ f $’s codomain and $ g $’s domain, these morphisms can be seamlessly composed.

The resulting morphism $ g \circ f $, mapping from $ A $ to $ C $, is detailed as follows:

$$ g \circ f: A \rightarrow C $$

$$ g(f(1)) = g(a) = x $$

$$ g(f(2)) = g(a) = x $$

$$ g(f(3)) = g(b) = y $$

In conclusion, morphism composition is a powerful tool in category theory, enabling the construction of new functions and relationships within a category, while preserving its structural integrity and operational rules.