Functors

Builds on

Required prerequisites, all the way down: 15

You will learn to

• Verify that the power-set, forgetful and hom functors are functors, and that a contravariant functor from C to D is the same as a functor Cᵒᵖ → D.
• Explain why functors preserve isomorphisms and use this to show that objects are not isomorphic.
• Describe the free monoid on a set (finite lists) and the free vector space on a set as functors.

Definition

• A functor F : C → D assigns to each object A of C an object F(A) of D and to each morphism f : A → B a morphism F(f) : F(A) → F(B), such that F(g ∘ f) = F(g) ∘ F(f) and F(id_A) = id_{F(A)}.
• A contravariant functor from C to D is a functor Cᵒᵖ → D.
• For an object A of a locally small category C, the hom functor Hom(A, −) : C → Set sends each object B to the set of morphisms A → B.

Theorem

• Functors preserve isomorphisms: if f is an isomorphism, so is F(f).
• Functors compose, and small categories with functors form a category Cat.
• The fundamental group is a functor from pointed spaces to groups; functoriality shows there is no retraction of the disc onto its boundary circle.

Common misconceptions

• Thinking a functor must be injective on objects or on morphisms.
• Believing every functor reflects isomorphisms, so that F(f) invertible would force f invertible; this fails in general.

Leads to

Optional discoveries

• Homotopy and the fundamental group π₁ is a functor, and functoriality gives a short proof of the Brouwer fixed-point theorem for the disc.
• The Yoneda lemma Hom functors are the raw material of the Yoneda lemma.
• ← Composition
• ← Algebraic structures: monoids, groups and rings
• ← Multivariable and vector calculus
• ← Groups
• ← Representation theory of finite groups
• ← Smooth manifolds

Learning pathways

• Undergraduate mathematics major

Further reading

• Tom Leinster, Basic Category Theory (2014)
• Emily Riehl, Category Theory in Context (2016)