EXERCISES IN CATEGORY THEORY 9
1. Adjoint functors
1.1. Algebra of adjoint functors.
(1) Consider U1 : A → B and U2 : B → C and adjunctions F1 U1 and F2 U2. Show
that we have a composite adjunction F1F2 U2U1.
(2) Show that the left adjoint of a functor U : A → B is unique up to natural isomorphism.
1.2. Adjoints and representables. Let C be a locally small category. Recall that a
functor U : C → Set is said to be representable if there exists a natural isomorphism
U ∼= C(X, −) for some X ∈ C.
(1) Let C and U be as above. Prove that if U has a left adjoint then U is representable.
(2) Assuming that C has all (infinite) coproducts show that the converse holds: U is
representable =⇒ U has a left adjoint.
1.3. Examples.
(1) We have seen that U : Mon → Set has a left adjoint (constructed using word
monoids). Prove that U does not have a right adjoint.
(2) Show that the forgetful functor U : Grp → Mon from groups to monoids does have a
right adjoint – construct it!
1.4. Further questions.
(1) Consider the forgetful functor O : Cat → Set which sends a small category to its set
of objects. Show that there is a string of four adjoints C D O I.
(2) Show that U : A → B has a left adjoint ⇐⇒ each category X/U has an initial object
⇐⇒ each functor B(X, U−) : A → Set is representable.1 2
(3) Given a category C a weakly initial set consists of a set of objects {Xi ∈ C, i ∈ I} such
that for each A ∈ C there exists some i ∈ I and a morphism Xi → A. Now given a
functor U : A → B we say that U satisfies the solution set condition if for each X ∈ B
the category X/U has a weakly initial set of objects. The general adjoint functor
theorem asserts that if C is a locally small category with limits then U : C → D has
a left adjoint if and only if U preserves limits and satisfies the solution set condition.
Use the adjoint functor theorem to verify that U : Grp → Mon (or your favourite
forgetful functor between algebraic categories) has a left adjoint.
(The key point here is to think about how to construct solution sets in algebraic cate-
gories.)
Date: November 12, 2014.
1Objects of the category X/U are pairs (f : X → UA, A). A morphism h : (f : X → UA, A) → (g :
X → UB, B) in X/U is given by a morphism h : A → B ∈ A such that Uh ◦ f = g.
2
The functor B(X, U−) : A → Set is the composite of U : A → B and B(X, −) : B → Set.
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