EXERCISES IN CATEGORY THEORY 8
1. Adjoint functors
1.1. Examples of adjoint functors. A functor U : A → B has a left adjoint if for each
X ∈ A there exists an object FX and morphism ηX : X → UFX with the following
universal property:
given A ∈ A and f : X → UA ∈ B there exists a unique arrow f : FX → A ∈ A such
that the triangle
UFX
f
##HHHHHHHHH
X
ηX
OO
f
// UY
(1.1)
commutes. Then FX is the value of the left adjoint to U.
(1) Let U : Mon → Set be the forgetful functor from monoids to sets. Given a set X
elements of the word monoid FX are lists [x1 . . . xn] of elements of X, with multiplication
given by joining lists: ie. [x, y][z] = [x, y, z]. Show that FX has the universal
property of the left adjoint to U.
(2) The forgetful functor U : CRing → Set from the category of commutative rings to the
category of sets has a left adjoint F. Show that the value of F at the 1-element set {x}
is the commutative ring of polynomials anxn
+ a1x + . . . a0 with integer coefficients
ai ∈ Z. What is FX where X is a finite set (or even an arbitrary set?)
(3) Consider U : V ect → Set. Show that the value of the left adjoint FX is the vector
space with basis set X.
(4) Consider the forgetful functor from topological spaces U : Top → Set to sets. Show
that the left adjoint to U sends a set X to X with the discrete topology: all subsets
are open.
(5) Given a set X let PX be the power set of X: since this is a poset we can view it as a
category. Given f : X → Y we get functors Pf : PX → PY : U → {fx ∈ Y : x ∈ U}
and f∗
: PY → PX : U → {x : fX ∈ U}. Show that Pf f∗
.
1.2. General categorical questions.
(1) Prove that given a collection of arrows as in (1.1) that the objects FX uniquely
give rise to a functor F : B → A such that the morphisms ηX : X → UFX are
the components of a natural transformation. In particular check that F preserves
composition.
(2) Prove that the left adjoint of a functor U : A → B is unique up to natural isomor-
phism.
Date: November 5, 2014.
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