Introduction My purpose in writing this exposition is to explain what adjoint functors are, prove a few of their properties, give a few examples, and to convince the reader that they are wonderful gems. Adjunctions elucidate the structure of mathematics by illuminating relationships between categories. In this section, I record any notation or results that I use later. • A 2 C means that A is an object of the category C. • Functors are sometimes named according to their formulae. In this case, the argument(s) are denoted with a dash: −. So the functor A 7! A ⊗ B for B fixed and A; B 2 C would be denoted − ⊗ B : C!C. By convention, when multiple dashes are used in the formula, they represent distinct arguments. • When discussing a hom functor (i.e. the functor Cop × C ! Set sending an ordered pair of objects to the set of morphisms pointing from the first to the second), it is useful to emphasize the category in which the objects live. For this reason, we will denote hom functors using the name of the category, so we have e.g. the hom functor C(−; −): Cop × C ! Set. Recall that the hom functor is functorial by pre and postcomposition in its two variables. •C D denotes the functor category of functors from D to C (with morphisms natural transformations). • For C a small category (i.e. one with a set, rather than a proper class, op of morphism), we identify PC def= SetC ; that is, PC is the category of presheaves on C. • A subcategory D ⊆ C is said to be full if for every A; B 2 D, f : A ! B a morphism of C, f is a morphism of D. • A presheaf on a category C is a functor Cop !Set. • We say that a presheaf F : Cop !Set is representable if it is isomorphic to a presheaf of the form C(−;A) for A 2 C. • Denote by Υ : C! PC the functor defined by A 7! C(−;A). This functor is known as the Yoneda embedding. • Let F : A!B, G; H : B!C, and I : C!D be functors, and let η : G ! H be a natural transformation. We define a new natural def transformation I ∗ η : IG ! IH by (I ∗ η)B = I(ηB) for B 2 B. Similarly, def we define a natural transformation η ∗ F : GF ! HF by (η ∗ F )A = ηF (A) for A 2 A. In the parlance of higher category theory, these constructions are known as \whiskering" η along I and F . 1 • Note that I do not explicitly articulate the duals to the results that I state. In particular, I've made the arbitrary choice to discuss left adjoints more than right adjoints. • I usually only define how functors act on objects. Technically, the definition of a functor requires a description of how it acts on morphisms as well. However, writers in this field tend to work from the assumption that the action of a functor on morphisms is obvious from its action on objects. Working out how a functor should act on morphisms given its action on objects is an excellent exercise for beginning category theorists. The following lemma and corollary will be used later in the paper and are important theorems of category theory, but they are also somewhat abstract, and may not feel intuitive at first blush. The reader should feel welcome to continue with the paper and return to these theorems later. Lemma (Yoneda) 1. For any presheaf F : Cop !Set and object A 2 C, there is an isomorphism F (A) =∼ PC(Υ(A);F ), and this isomorphism is natural in both F and A. This is a generalization of Cayley's Theorem when we consider groups as single-object categories. Proof. We can see this by considering idA 2 Υ(A). Given a natural transfor- mation f : Υ(A) ! F and a morphism g : B ! A in C, we have by naturality that fB(g) = F (g)(fA(idA)), so fA(idA) completely determines the natural transformation, and the function fA has codomain F (A). Corollary 2. The Yoneda embedding lives up to its name: it embeds C as a full subcategory of PC, identifying C with the subcategory of representable presheaves on C. Proof. PC(Υ(A); Υ(B)) =∼ Υ(B)(A) = C(A; B). Adjunctions Definition 3. A functor L : C!D is said to be left adjoint to a functor R : D!C if there exists a natural isomorphism D(L(−); −) =∼ C(−;R(−)). Note that these are functors Cop × D ! Set. In this case, L is a left adjoint functor.A right adjoint functor is the dual notion, so a functor is a left adjoint if it has a right adjoint and vice versa. We use the notation L a R to express \L is left adjoint to R". The reader should verify the following elementary examples. Example 4. Let C be a category with a terminal object and D be a category with an initial object. Let L : C!D be the constant functor defined by L : A 7! ; for all A 2 C and ; initial in D. Let R : D!C be the constant functor defined by R : B 7! ∗ for any B 2 D and ∗ terminal in C. Then L a R. 2 Example 5. Let F : Set !Grp be the functor sending a set to the free group it generates, and let U : Grp !Set be the functor sending a group to its underlying set. Then F a U. It is useful when reading mathematical texts that use category theory know that that when an object of some category is referred to as \free", this almost always means that it is in the essential image of some left adjoint functor (\in the essential image" means that the object is isomorphic to some object in the image). Example 6. Let Grph be the category of multi digraphs allowing vertices to be (multiply) self-adjacent. Let the functor U : Cat !Grph turn a small category into a multi digraph by taking the objects to be vertices, the arrows to be directed edges, and forgetting composition. U has a left adjoint which sends a graph to the free category it generates: the objects are the vertices, each edge determines an morphism, identity morphisms are added in (i.e. they are not presumed to be equal to the morphism determined by an edge witnessing a self-adjacency), and any finite sequence of \composable" edges determines a (new) morphism, with composition determined by concatenation of strings of \composable" edges. There is a similar adjunction between the category of small groupoids and the category of undirected multigraphs. Example 7. The functor U : T op !Set sending a topological space to its set of points has both a left and a right adjoint. The left adjoint sends a set to the discrete space with that set of points, while the right adjoint sends a set to the indiscrete space with that set of points. Example 8. For S a set, denote its powerset PS (there is a precise sense in which this notation agrees with the previously established notation for presheaf categories; perhaps I will discuss this further in a sequel note if there is interest in one, or certainly in person if anyone would like). We can regard PS as a category by positing the existence of a unique element of PS(U; V ) iff U ⊆ V . Then a function f : S ! T determines a functor f! : PS ! PT sending a subset to its image. We also have a functor f ∗ : PT ! PS sending a subset to its preimage. ∗ ∗ f! a f . But perhaps less obviously, f has a right adjoint as well. This is c c c the functor f∗ : PS ! PT defined by f∗ : U 7! (f(U )) where U denotes the complement of U. Example 9. Unsurprisingly, adjunctions arise in the context of other category- theoretic constructions, and often prove to be useful tools in these instances. Let C be a category with finite coproducts. For an object X 2 C, the under category, denoted C=X , is the category with objects morphisms X ! A for A 2 C and morphisms f : g ! h given by commutative triangles X g h f A B 3 f There is a forgetful functor CX= !C given by (X ! A) 7! A, and this functor has a left adjoint given by A 7! (X ! X t A). When C is T op and X is ∗, the point, T op∗= is the category of pointed spaces, and the left adjoint just entails adjoining a disjoint basepoint. Example 10. Let X and Y be spaces and let ShX and ShY be the categories of sheaves on X and Y respectively. Then a continuous map f : X ! Y determines a pushforward functor f∗ : ShX !ShY which is left adjoint to the pullback ∗ functor f : ShY !ShX . Proposition 11. A left adjoint functor has a unique right adjoint up to unique natural isomorphism. Proof. Suppose L a R and L a R0. Then for A 2 C;B 2 D, C(A; R(B)) =∼ D(L(A);B) =∼ C(A; R0(B)), so by the Yoneda Lemma, R(B) =∼ R0(B) and this isomorphism is natural in B, so it extends to a natural isomorphism R =∼ R0. Proposition 12. The composition of two left adjoint functors is a left adjoint. Proof. Take L1 : C!D, L2 : D!E with right adjoints R1, R2 respectively. ∼ ∼ Then E(L2L1A; B) = D(L1A; R2B) = C(A; R1R2B).