Mathematics.

maps between categories

Functors

Category Theory80 minDifficulty7 out of 10

You should know: categories and morphisms

Overview

A functor is a structure-preserving map between categories. Just as a group homomorphism sends elements to elements while preserving the group operation, a functor sends objects to objects and morphisms to morphisms while preserving composition and identities. Functors are the morphisms of the category \(\text{Cat}\) of all small categories, making category theory itself categorical. Concrete examples appear everywhere: the fundamental group \(\pi_1\), homology groups \(H_n\), and forgetful functors from algebraic to set-theoretic categories are all functors.

Intuition

A functor is a 'translation dictionary' between two mathematical worlds. It converts every object in the source category to an object in the target, and every arrow to an arrow, in a way that respects how arrows combine. A contravariant functor reverses the direction of arrows — like the operation of taking dual vector spaces, which turns a linear map \(T: V \to W\) into its transpose \(T^*: W^* \to V^*\).

Formal Definition

Definition

A covariant functor \(F: \mathcal{C} \to \mathcal{D}\) consists of: (1) an object function \(F: \text{Ob}(\mathcal{C}) \to \text{Ob}(\mathcal{D})\); (2) for each pair \(A, B \in \mathcal{C}\), a morphism function \(F: \text{Hom}_{\mathcal{C}}(A,B) \to \text{Hom}_{\mathcal{D}}(FA, FB)\); satisfying functoriality:

F(idA)=idFAfor all A∈CF(\text{id}_A) = \text{id}_{FA} \quad \text{for all } A \in \mathcal{C}

Functors preserve identity morphisms

functor-identity
F(g∘f)=F(g)∘F(f)whenever f:A→B,  g:B→CF(g \circ f) = F(g) \circ F(f) \quad \text{whenever } f: A \to B,\; g: B \to C

Functors preserve composition

functor-composition
A contravariant functor F:C→D satisfies F(g∘f)=F(f)∘F(g)\text{A contravariant functor } F: \mathcal{C} \to \mathcal{D} \text{ satisfies } F(g \circ f) = F(f) \circ F(g)

Contravariant functors reverse the order of composition

contravariant

Notation

NotationMeaning
F:C→DF: \mathcal{C} \to \mathcal{D}F is a functor from C to D
Ff or F(f)Ff \text{ or } F(f)The image of morphism f under functor F
FA or F(A)FA \text{ or } F(A)The image of object A under functor F
Cop\mathcal{C}^{\text{op}}The opposite category of C (same objects, reversed morphisms)
Hom(A,−)\text{Hom}(A, -)Covariant hom-functor sending B to Hom(A,B)
Hom(−,B)\text{Hom}(-, B)Contravariant hom-functor sending A to Hom(A,B)

Properties

Faithful functor

F:C→DisfaithfulifeachfunctionFA,B:Hom(A,B)→Hom(FA,FB)isinjective.F: \mathcal{C} \to \mathcal{D} is faithful if each function F_{A,B}: \text{Hom}(A,B) \to \text{Hom}(FA,FB) is injective.

Example: The forgetful functor \(U: \text{Grp} \to \text{Set}\) is faithful: distinct group homomorphisms give distinct functions.

Full functor

FisfullifeachFA,Bissurjective.F is full if each F_{A,B} is surjective.

Example: The inclusion functor \(\text{Ab} \hookrightarrow \text{Grp}\) is full: every group homomorphism between abelian groups is already a map in Ab.

Essentially surjective

FisessentiallysurjectiveifforeveryD∈DthereexistsC∈CwithFC≅D.F is essentially surjective if for every D \in \mathcal{D} there exists C \in \mathcal{C} with FC \cong D.

Condition: Together with full and faithful, this characterizes equivalences of categories.

Equivalence of categories

AfunctorF:C→Disanequivalenceifitisfull,faithful,andessentiallysurjective.A functor F: \mathcal{C} \to \mathcal{D} is an equivalence if it is full, faithful, and essentially surjective.

Example: The category of finite-dimensional real vector spaces is equivalent to the category whose objects are natural numbers \(n\) and morphisms are \(n \times m\) matrices.

Worked Examples

  1. Objects: A pointed space \((X, x_0)\) maps to the fundamental group \(\pi_1(X, x_0)\).

    π1(X,x0)={[γ]∣γ:[0,1]→X,  γ(0)=γ(1)=x0}\pi_1(X, x_0) = \{ [\gamma] \mid \gamma: [0,1] \to X,\; \gamma(0)=\gamma(1)=x_0 \}

  2. Morphisms: A pointed continuous map \(f: (X, x_0) \to (Y, y_0)\) induces a group homomorphism \(f_*: \pi_1(X,x_0) \to \pi_1(Y,y_0)\) by post-composition: \(f_*([ \gamma]) = [f \circ \gamma]\).

    f∗([γ])=[f∘γ]f_*([\gamma]) = [f \circ \gamma]

  3. Identity preservation: The identity map \(\text{id}: (X,x_0) \to (X,x_0)\) induces \((\text{id})_*([ \gamma]) = [\text{id} \circ \gamma] = [\gamma]\), so \((\text{id})_* = \text{id}_{\pi_1(X)}\).

  4. Composition preservation: For \(f: (X,x_0) \to (Y,y_0)\) and \(g: (Y,y_0) \to (Z,z_0)\):

    (g∘f)∗([γ])=[(g∘f)∘γ]=[g∘(f∘γ)]=g∗(f∗([γ]))=(g∗∘f∗)([γ])(g \circ f)_*([\gamma]) = [(g \circ f) \circ \gamma] = [g \circ (f \circ \gamma)] = g_*(f_*([\gamma])) = (g_* \circ f_*)([\gamma])

  5. Thus \(\pi_1\) is a covariant functor from \(\text{Top}_*\) to \(\text{Grp}\).

Answer: \(\pi_1\) is a covariant functor: it sends pointed spaces to groups and pointed continuous maps to group homomorphisms, preserving identities and composition.

Practice Problems

Difficulty 6/10

Prove that the forgetful functor \(U: \text{Grp} \to \text{Set}\) (sending a group to its underlying set and a group homomorphism to the underlying function) is faithful but not full.

Difficulty 7/10

Describe the covariant hom-functor \(\text{Hom}(\mathbb{Z}, -): \text{Ab} \to \text{Ab}\) and identify what it does to objects and morphisms. What group does it send \(A\) to?

Difficulty 8/10

A functor \(F: \mathcal{C} \to \mathcal{D}\) that is full and faithful reflects isomorphisms: if \(Ff\) is an isomorphism in \(\mathcal{D}\), prove that \(f\) is an isomorphism in \(\mathcal{C}\).

Common Mistakes

Common Mistake

A functor that sends every morphism to an identity is a valid functor.

The constant functor (sending all morphisms to the identity) only works if all composites in the image coincide, which fails unless the target category is trivial. The constant functor on objects is fine, but morphisms must be sent consistently.

Common Mistake

Full and faithful implies an isomorphism of categories.

Full and faithful (without essentially surjective) only gives a functor that is injective on morphisms and surjective on hom-sets. Essentially surjective is additionally needed for an equivalence of categories. Even then, an equivalence need not be an isomorphism (bijection on objects).

Common Mistake

A contravariant functor can be composed with a covariant functor in any order.

The variance of the composite depends on the order. Contra ∘ Covariant = Contravariant. Contra ∘ Contra = Covariant. Track variances carefully, or use opposite categories to reduce everything to covariant functors.

Quiz

Which property does a functor NOT necessarily preserve?
The functor \(\text{Hom}(-, B): \mathcal{C}^{\text{op}} \to \text{Set}\) is:
An equivalence of categories requires the functor to be:
The composite of two contravariant functors \(F: \mathcal{C} \to \mathcal{D}\) and \(G: \mathcal{D} \to \mathcal{E}\) (both contravariant) is:

Summary

  • A functor \(F: \mathcal{C} \to \mathcal{D}\) assigns to each object and morphism of \(\mathcal{C}\) an object and morphism of \(\mathcal{D}\), preserving identities and composition.
  • Covariant functors preserve arrow direction; contravariant functors reverse it (equivalently, they are covariant functors on the opposite category).
  • Key examples: \(\pi_1\) (fundamental group), \(H_n\) (homology), dual space \((-)^*\), forgetful functors, and free functors.
  • A functor is faithful/full/essentially surjective if its action on hom-sets is injective/surjective/hits all objects up to isomorphism; all three together give an equivalence of categories.
  • The composite of two contravariant functors is covariant; functors compose associatively, making \(\text{Cat}\) itself a category.

References

  1. BookMac Lane, S. Categories for the Working Mathematician, 2nd ed. Springer, 1998. Ch. I.
  2. BookAwodey, S. Category Theory, 2nd ed. Oxford University Press, 2010. Ch. 2.
  3. WebsiteWikipedia — Functor
  4. WebsitenLab — functor