Categories and Functors - Core Definitions

Category theory provides a unified framework for studying mathematical structures and their relationships. At its foundation lies the concept of a category, which abstracts the notion of objects and morphisms between them.

DefinitionCategory

A category $\mathcal{C}$ consists of:

A collection $\text{Ob}(\mathcal{C})$ of objects
For each pair of objects $A, B$ , a collection $\text{Hom}_\mathcal{C}(A, B)$ of morphisms (or arrows) from $A$ to $B$
For each object $A$ , an identity morphism $\text{id}_A \in \text{Hom}_\mathcal{C}(A, A)$
A composition operation that assigns to morphisms $f \in \text{Hom}_\mathcal{C}(A, B)$ and $g \in \text{Hom}_\mathcal{C}(B, C)$ a composite morphism $g \circ f \in \text{Hom}_\mathcal{C}(A, C)$

These must satisfy:

Associativity: $(h \circ g) \circ f = h \circ (g \circ f)$ whenever the compositions are defined
Identity laws: For any $f \in \text{Hom}_\mathcal{C}(A, B)$ , we have $f \circ \text{id}_A = f = \text{id}_B \circ f$

ExampleFundamental Categories

Important examples of categories include:

Set: Objects are sets, morphisms are functions
Grp: Objects are groups, morphisms are group homomorphisms
Top: Objects are topological spaces, morphisms are continuous maps
Vect $_k$ : Objects are vector spaces over field $k$ , morphisms are linear transformations
Poset: A partially ordered set $(P, \leq)$ forms a category with objects being elements of $P$ and a unique morphism $x \to y$ whenever $x \leq y$

DefinitionFunctor

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A functor $F: \mathcal{C} \to \mathcal{D}$ consists of:

An object function that assigns to each object $A$ in $\mathcal{C}$ an object $F(A)$ in $\mathcal{D}$
A morphism function that assigns to each morphism $f: A \to B$ in $\mathcal{C}$ a morphism $F(f): F(A) \to F(B)$ in $\mathcal{D}$

These must satisfy:

Identity preservation: $F(\text{id}_A) = \text{id}_{F(A)}$ for all objects $A$
Composition preservation: $F(g \circ f) = F(g) \circ F(f)$ for all composable morphisms $f, g$

Functors can be thought of as structure-preserving maps between categories. They relate different mathematical contexts while maintaining the categorical structure.

ExampleForgetful Functor

The forgetful functor $U: \textbf{Grp} \to \textbf{Set}$ sends each group to its underlying set and each group homomorphism to its underlying function. This functor "forgets" the group structure while preserving the set-theoretic data.

Similarly, the functor $U: \textbf{Vect}_k \to \textbf{Set}$ forgets the vector space structure, sending each vector space to its underlying set of vectors.

DefinitionContravariant Functor

A contravariant functor $F: \mathcal{C} \to \mathcal{D}$ reverses the direction of morphisms. It assigns:

To each object $A$ in $\mathcal{C}$ , an object $F(A)$ in $\mathcal{D}$
To each morphism $f: A \to B$ in $\mathcal{C}$ , a morphism $F(f): F(B) \to F(A)$ in $\mathcal{D}$ (note the reversal)

The composition axiom becomes $F(g \circ f) = F(f) \circ F(g)$ .

A standard (covariant) functor is sometimes called a covariant functor for emphasis.

Remark

The distinction between covariant and contravariant functors is fundamental in category theory. Many natural constructions, such as taking dual spaces in linear algebra or computing cohomology groups in topology, are naturally contravariant.

The language of categories and functors provides a powerful abstraction that reveals deep connections between seemingly different areas of mathematics. By focusing on objects, morphisms, and structure-preserving maps, category theory enables mathematicians to recognize and exploit common patterns across diverse mathematical disciplines.