2.3 Composition and identity morphisms

Composition and identity morphisms are the building blocks of category theory. They allow us to connect objects and study their relationships, forming the foundation for more complex structures.

These concepts are essential for understanding how categories work. Composition lets us chain morphisms together, while identity morphisms act as neutral elements, ensuring consistent behavior within the category.

Composition and Identity Morphisms in Category Theory

Role of composition in categories

• Composition fundamental operation combines morphisms to create new morphisms between objects
• Given morphisms $f: A \to B$ and $g: B \to C$, composition yields morphism $g \circ f: A \to C$ (function composition, group homomorphisms)
• Composition allows chaining together morphisms to study relationships between objects in a category (paths in a directed graph)
• Composition binary operation on set of morphisms takes two morphisms as input and produces single morphism as output
• Existence of composition one of defining properties of a category enables study of how objects relate through morphisms

Associativity and identity laws

• Associativity of composition: $(h \circ g) \circ f = h \circ (g \circ f)$ order of composition does not affect result (matrix multiplication)
• Left identity law: $id_B \circ f = f$ for any morphism $f: A \to B$ composing with identity on codomain yields original morphism
• Right identity law: $f \circ id_A = f$ for any morphism $f: A \to B$ composing with identity on domain yields original morphism
• Identity laws ensure composition behaves consistently and predictably in a category (neutral element in arithmetic)

Importance of identity morphisms

• Identity morphisms special morphisms exist for every object in a category (identity function, identity matrix)
• For each object $A$, unique morphism $id_A: A \to A$ called identity morphism on $A$
• Identity morphisms act as neutral elements with respect to composition (multiplicative identity in rings)
• Composing any morphism with appropriate identity morphism yields original morphism
• Existence of identity morphisms another defining property of a category ensures composition behaves correctly
• Identity morphisms allow objects to be mapped to themselves (self-loops in a graph)
• Identity morphisms crucial for formulating identity laws essential for category structure

Examples of composition and identities

• In category of sets (Set):
  • Morphisms are functions between sets
  • Composition is usual function composition: $(g \circ f)(x) = g(f(x))$
  • Identity morphism on set $A$ is identity function $id_A(x) = x$ for all $x \in A$
• In category of groups (Grp):
  • Morphisms are group homomorphisms
  • Composition is composition of group homomorphisms
  • Identity morphism on group $G$ is identity homomorphism $id_G: G \to G$ defined by $id_G(g) = g$ for all $g \in G$
• In category of topological spaces (Top):
  • Morphisms are continuous functions
  • Composition is composition of continuous functions
  • Identity morphism on topological space $X$ is identity function $id_X: X \to X$ defined by $id_X(x) = x$ for all $x \in X$
• In category of vector spaces (Vect):
  • Morphisms are linear transformations
  • Composition is composition of linear transformations
  • Identity morphism on vector space $V$ is identity transformation $id_V: V \to V$ defined by $id_V(v) = v$ for all $v \in V$