sign in sign up
<<
Home , Adjoint functors, Category (mathematics), Cokernel, Functor, Functor category, Hom functor

Introduction

My purpose in writing this exposition is to explain what adjoint 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 by illuminating relationships between categories. In this , I record any notation or results that I use later.

• A ∈ C means that A is an object of the C. • Functors are sometimes named according to their formulae. In this case, the argument(s) are denoted with a dash: −. So the A 7→ A ⊗ B for B fixed and A, B ∈ C would be denoted − ⊗ B : C → C. By convention, when multiple dashes are used in the formula, they represent distinct arguments. • When discussing a (i.e. the functor Cop × C → sending an of objects to the set of 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 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 , op of ), we identify PC def= SetC ; that is, PC is the category of presheaves on C. • A D ⊆ C is said to be full if for every A, B ∈ 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 ∈ C. • Denote by Υ : C → PC the functor defined by A 7→ C(−,A). This functor is known as the Yoneda .

• Let F : A → B, G, H : B → C, and I : C → D be functors, and let η : G → H be a . We define a new natural def transformation I ∗ η : IG → IH by (I ∗ η)B = I(ηB) for B ∈ B. Similarly, def we define a natural transformation η ∗ F : GF → HF by (η ∗ F )A = ηF (A) for A ∈ A. In the parlance of higher , 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 ∈ C, there is an 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 ∈ 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 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 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 ∈ C and ∅ initial in D. Let R : D → C be the constant functor defined by R : B 7→ ∗ for any B ∈ 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 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 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 to the it generates: the objects are the vertices, each edge determines an morphism, 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 and the category of undirected . Example 7. The functor U : T op → Set sending a topological to its set of points has both a left and a right adjoint. The left adjoint sends a set to the 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 . For an object X ∈ C, the under category, denoted C/X , is the category with objects morphisms X → A for A ∈ C and morphisms f : g → h given by commutative triangles

X g h

f A B

3 f There is a 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 f : X → Y determines a 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 ∈ C,B ∈ D, C(A, R(B)) =∼ D(L(A),B) =∼ C(A, R0(B)), so by the , 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 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). Example 13. Let I : C → CD be the functor sending an object A of C to the constant functor D → C taking the value A. Its left adjoint, if it exists, is the functor that sends a functor D → C to its colimit in C. Proposition 14. Let L : C → E be a left adjoint functor and F : D → C be a functor (often called a diagram in this context). Then colim LF =∼ L(colim F ) as long as the latter exists. That is, left adjoint functors perseve (or commute with) colimits.

D D Proof. Let I1 : E → E and I2 : C → C be the functors sending objects to the corresponding constant diagrams as in Example 13. Let R : E → C be a right adjoint functor, and let RD : ED → CD be the functor defined by F 7→ RF , i.e. ∼ D which applies R to diagrams pointwise. I1R = R I2, so the result follows by Example 13, Proposition 12, and Proposition 11. Example 15. Fix a commutative R. Let C be the category whose objects are triples (A, f, g) such that A is a , f : R → A, g : A → R, and 0 0 gf = idR, and whose morphisms ϕ :(A, f, g) → (B, f , g ) are ring morphisms ϕ : A → B such that ϕf = f 0 and g0ϕ = g. Let D be the category of R-modules. Then given an object (A, f, g) of C, the of g is an of A, which has the structure of an R- via f. This determines a functor C → D. This functor has a left adjoint given by M 7→ Sym(M), the functor sending a module to its . The required map Sym(M) → R is the one fixing elements of R ⊂ Sym(M) and sending elements M ⊂ Sym(M) to 0 ∈ R.

4 The following is an equivalent definition of an adjunction. Although it is equivalent in the context of adjoint functors between categories, it is the correct way to extend the definition of adjunction to abstract 2-categories, which will not be discussed further in this section. This definition may at first seem abstract, but I will clarify it with examples shortly. Theorem 16. A left adjoint functor is a functor L : C → D such that there exists a functor R : D → C and natural transformations η : idC → RL (called the ) and  : LR → idD (the counit) such that ( ∗ L) ◦ (L ∗ η) = idL and (R ∗ ) ◦ (η ∗ R) = idR (◦ here denotes composition of natural transformations; this would normally be denoted by juxtaposition, be we include the notation for readability). Proof. A proper, fully detailed and explicit treatment of this theorem is worth reading, but laborious to write. See Proposition 4.2.6 on page 124 of Emily Riehl’s Category Theory in Context. However, in case you do not look up the reference, I will at least indicate the construction without verifying the properties. Given an adjunction according C ∼ to our first definition, the unit η : idC → RL is the image of idL ∈ D (L, L) = C C (idC,RL) and dually the counit. In the opposite direction, given an adjunction according to this new definition, we define a function natural in C and D D(LC, D) → C(C,RD) by f 7→ Rf ◦ ηC ; this is the desired natural isomorphism, and its inverse uses postcomposition by the counit rather than precomposition by the unit.

Example 17. We use the notation of Example5. The unit morphism ηS : S → UFS sends an element of S to the element it generates in the underlying set of the on S, and the counit G : FUG → G sends a of the free group FUG to the element of G that generated it (and the behavior of the morphism on the rest of the group FUG is determined by the fact that G is a group ). We strongly encourage the reader to consider the units and counits of the other examples of adjunctions we presented earlier. Definition 18. A full subcategory D ⊆ C is said to be reflective if the inclusion functor is a right adjoint. It is an exercise to check that for adjunctions arising from reflective subcate- gories, the counit is a natural ismorphism. Let D be a reflective subcategory of C and L the left adjoint to the inclusion functor. We can further verify that for any η C ∈ C and any D ∈ D, any morphism C → D factors uniquely as C →C LC → D. Often, D is the subcategory of objects satisfying some extra property (alright, this is redundant, since “membership in D” is a property), and the unit connects an object to its best approximation in D. Example 19. The category Ab of abelian groups is a reflective subcategory of the category Grp of groups. More generally, the category CMon of commutative is a reflective subcategory of the category Mon of monoids.

5 Example 20. The category T or of torsion abelian groups is a coreflective subcategory of Ab, and the category NT or of torsion-free abelian groups is a reflective subcategory of Ab. The counit of the former adjunction is the kernel of the unit of the latter adjunction (and dually, the unit of the latter adjunction is the of the counit of the former adjunction, so that we obtain a short of functors that splits object-wise).

Example 21. Grp is a reflective and a coreflective subcategory of Mon. The left adjoint to the inclusion (called the group completion of a ) freely adjoins an inverse to each element of a given monoid and then makes the necessary identifications for the resulting to be well-defined. The right adjoint to the inclusion simply strips away all non-invertible elements from a given monoid. Example 22. Let HCom ⊂ T op be the full subcategory of compact Hausdorff spaces in the category of topological spaces. HCom is reflective, and the left adjoint to the inclusion is given by Stone-Cechˇ compactification. I’ll conclude with a quasi-example of an adjuction that is undone by size issues. I will not provide a proof of the claim. Even so, it is a fun example of category theory applied to category theory. Almost Example 23. Let Cat be the category of small categories and functors, and let Pres be the category of locally presentable categories with left adjoint functors its morphisms. Don’t worry too much what a locally presentable category is, but know that for a small category C, PC is always locally presentable (and it’s nice to know that locally presentable categories are by definition cocomplete: that is, they admit all small colimits). Given a functor F : C → D in Cat, we obtain a functor F ∗ : PD → PC defined ∗ by g 7→ gF . F always has a left adjoint denoted F! (the construction uses something called “coends” and is beyond the scope of this exposition). Then we have a functor L : Cat → Pres sending a small category C ∈ Cat to PC and a functor F : C → D to F! : PC → PD. If there existed an inclusion functor I : Pres → Cat, we would have L a I, and the Yoneda embedding would be the unit of the adjunction! But we do not have such a functor I because Pres contains large categories, while Cat contains only small ones. Why not let Cat be the category of all categories? Well, this is problematic for evident set-theoretic reasons. The proper way to salvage this situation is as follows. We posit the existence of two inaccessible cardinals κ0 < κ1. Then we’ll say that a category is tiny if it has a set (rather than a proper class) of morphisms and that set has cardinality less than κ0, and that a category is small if it has a set of morphisms with cardinality less than κ1. Let Cat0 be the category of tiny categories, let Pres be the category of small locally presentable categories, and let Cat1 be the category of small categories. Then we have a functor L : Cat0 → Pres as above, an inclusion functor IP : Pres → Cat1, and an inclusion functor I0 : Cat0 → Cat1. Then the Yoneda embedding is a natural transformation Υ : I0 → IP L, and

6 this natural transformation satisfies a very similar to the one satisfied by the unit of an adjunction: given C ∈ Cat0, D ∈ Pres, and F : I0(C) → IP (D), F factors as

ΥC IP (G) I0(C) IP L(C) IP (D) for a functor G : L(C) → D unique up to natural ismorphism. It follows that Pres(L(C), D) 'Cat1(I0(C),IP (D)), where ' denotes not isomorphism but equivalence of cateogires, a state of affairs also very reminiscent of an adjunction. Anyway, the lesson you should take from all this is that PC is the category obtained by freely adjoining all small colimits to C. Alternatively, it is the free locally presentable category generated by C. My aim with this exposition has not so much been to teach about adjunctions, but to advertise them (of course, the latter is impossible without some of the former). Still, I hope you’ve learned something, and that that something has sparked an interest. If so, let me know! I love to talk about category theory. If I were to make a sequel to this, the most natural topic to me would be theory, but other fun topics in category theory include (co)monads, their categories of (co)algebras, and (co)monadic adjunctions; 2- and 3-category theory; , the study of abelian categories, a particular class of enriched categories; theory, a field important to and logic which studies categories with formal similarities to the ; and the homotopical approach to , i.e. (∞, 1)-categories and (∞, n)-categories and their models, such as quasicategories and Θn-spaces. I have not the time to write a sequel now, but I’d love to talk to anyone about these subjects.

7