EXERCISES IN CATEGORY THEORY 7
1. Equivalence of categories
An equivalence of categories consists of categories C and D together with functors
F : C → D and G : D → C and natural isomorphisms 1C → GF and 1D → FG.1
We also
call the functors involved F : C → D and D → C equivalences of categories.
(1) Let Cat1 denote the category of categories with one object and functors between
them. There is a functor U : Cat1 → Mon which sends a category C with one object
x to the monoid C(x, x). Show that U is part of an equivalence of categories.
(2) Show that equivalence of categories is an equivalence relation on categories.
A functor U : C → D is said to be essentially surjective if given a ∈ D there exists b ∈ C
such that Fb is isomorphic to a in D. A functor U : C → D is said to be fully faithful if
given f : Ua → Ub there exists a unique arrow g : a → b such that Ug = f. (In other
words, the function Fa,b : C(a, b) → D(Fa, Fb) is a bijection.)
In fact a functor U : C → D is an equivalence of categories if and only if it is essentially
surjective and fully faithful.
Use this result where useful in the following exercises.
(3) Setpar is the category of sets and partial functions. A partial function (U, f) : X → Y
consists of a subset U ⊆ X and function f : U → Y . Thus a function from X to Y
which is only defined on U. The composite of (U, f) : X → Y and (V, g) : Y → Z is
the partial function (W, gf) : X → Z where W = {x ∈ U : fx ∈ V }. (Draw some
diagrams to see this).
The category of pointed sets Set∗ has objects (X, x) where x ∈ X and morphisms
f : (X, x) → (Y, y) are functions such that fx = y.
There is a functor F : Setpar → Set∗ defined by FX = (X + 1, ∗ ∈ 1) and sends
(U, f) : X → Y to k : (X + 1, ∗) → (Y + 1, ∗) defined by kx = fx if x ∈ U and
k(y) = ∗ otherwise. Show that F is a functor and an equivalence of categories.
(4) Show that if F : C → D is an equivalence then it preserves terminal and initial objects.
(In fact, an equivalence preserves any limits or colimits that exist.)
(5) (Harder) Prove that a functor is an equivalence of categories if and only if it is
essentially surjective on objects and fully faithful.
Date: October 29, 2014.
1Note that the direction of the natural isomorphisms does not matter since if 1C → GF is a natural
isomorphism then we get a natural isomorphism GF → 1C by taking inverses.
1