Categories, Functors, and Natural Transformations

Categories, functors, and natural transformations are the three basic notions of category theory, formalizing structures, the maps between structures, and the maps between such maps.

Definition

A category consists of objects and morphisms that compose associatively with identities; a functor maps objects and morphisms of one category to another preserving composition and identities; a natural transformation assigns to each object a morphism so that it commutes with the actions of two functors.

Scope

This topic covers the definition of a category by objects, morphisms, composition, and identities, the notion of a functor as a structure-preserving map between categories, natural transformations as morphisms of functors, and the resulting notions of isomorphism, equivalence of categories, and the Yoneda embedding.

Core questions

• What data and axioms define a category?
• How does a functor transport structure from one category to another?
• What does naturality mean and why is it the right notion of map between functors?
• When are two categories equivalent rather than equal?

Key theories

Category and functor axioms

Composition of morphisms is associative and unital, and functors preserve this compositional structure, so categorical constructions are stable under the maps that relate categories.

Natural transformations

A natural transformation relates two functors by a family of morphisms compatible with all the maps in the source category, capturing the informal idea of a construction defined uniformly and without arbitrary choices.

Yoneda lemma and embedding

Natural transformations out of a represented functor correspond to elements, so every object is determined by its morphisms and embeds fully and faithfully into a functor category.

Clinical relevance

These three notions are the vocabulary in which categorical mathematics is written: functors formalize constructions such as forming a fundamental group or a polynomial ring, naturality identifies canonical constructions, and the Yoneda perspective grounds the structural view that pervades algebra, topology, and the semantics of programming languages.

History

Eilenberg and Mac Lane introduced categories, functors, and natural transformations in 1945, with natural transformations as the motivating concept that required the others to be defined precisely. The Yoneda lemma, attributed to Nobuo Yoneda, soon became the cornerstone expressing the representability viewpoint of the subject.

Key figures

• Samuel Eilenberg
• Saunders Mac Lane
• Nobuo Yoneda

Seminal works

• maclane1998
• awodey2010
• riehl2016

Frequently asked questions

What is the point of natural transformations?

They make precise when a construction is canonical, defined the same way for every object without arbitrary choices. The classic example is the natural map from a vector space to its double dual, which exists uniformly, unlike the map to the single dual, which depends on a choice of basis.

What is an equivalence of categories?

It is a pair of functors between two categories whose composites are naturally isomorphic to the identities. Equivalent categories share all categorical properties even when they are not literally identical, which is the appropriate notion of sameness in category theory.