Perry Hart K-theory reading seminar UPenn September 26, 2018 Abstract We introduce the concept of a natural transformation in category theory. Afterward, we describe equivalences and adjunctions. The main sources for this talk are the following. • nLab. • John Rognes’s Lecture Notes on Algebraic K-Theory, Ch. 3. • Peter Johnstone’s lecture notes for “Category Theory” (Mathematical Tripos Part III, Michaelmas 2015), Ch. 1. 1 Natural transformations Let C and D be categories and F and G be functors C → D.A natural transformation φ : F ⇒ G is a function A 7→ fA from ob C to mor D such that fA is a map F (A) → G(A) and the following diagram commutes for any morphism h : A → B in C . FA F h FB fA fB GA GB Gh In symbols, this may be written as fBh∗ = h∗fA, where fA is called a component of φ. −1 Note 1.1. If every fA is an isomorphism, then the maps (fA) define a natural transformation G ⇒ F . If each fA is an isomorphism, then we say that φ is a natural isomorphism. Note that if D is a groupoid (i.e., a category in which every morphism is an isomorphism), then φ must be a natural isomorphism. Let F , G, and H be functors C → D. The identity natural transformation IdF : F ⇒ F is given by A 7→ IdF (A). Moreover, given natural transformations φ : F → G and ψ : G → H, define the composite natural transformation ψ ◦ φ by A 7→ (ψ ◦ φ)A := ψA ◦ φA. Lemma 1.2. A natural transformation φ : F ⇒ G is a natural isomorphism iff it has an inverse φ−1 : G ⇒ F . Proof. This follows from Note 1.1 along with our definition of a composite natural transformation. Example 1.3. 1. Let R and S be commutative rings. Any ring homomorphism f : R → S induces a ring homomorphism GLn(f) : GLn(R) → GLn(S) satisfying f(det(A)) = det GL(f)(A) . n 1 ∗ ∗ By viewing GLn and R 7→ R as functors from Ring to Grp and detR : GLn(R) → R as a mor- ∗ ∗ phism in Grp, we see that detR defines a natural transformation φ : GLn ⇒ f where f denotes ∗ ∗ f R∗ : R → S . GLn(f) GLn(R) GLn(S) detR detS R∗ S∗ f ∗ 2. Consider the power set functor P : Set → Set defined on objects by A 7→ P(A) and on morphisms g by Pg(S) = g(S). Then the function fA : A → P(A) given by a 7→ {a} defines a natural transformation φ : IdSet ⇒ P. ab 3. Set C = D = Grp, F = IdC , and G = (−) . Then given a group H, the natural projection ab f : H H induces a natural transformation φ : F ⇒ G. 4. We can view preorders (P, ≤) and (Q, ≤) as small categories and functors F, G : P → Q as order- preserving functions. Then there is a unique natural transformation φ : F ⇒ G iff F (x) ≤ G(x) for every x ∈ P . 5. The inversion isomorphism from a group G to its opposite group Gop defines a natural transformation op op φ : IdGrp ⇒ ((−) : Grp → Grp). In this sense, G is naturally isomorphic to G . Definition 1.4. Let C and D be categories with C small. The functor category Fun(C , D) := D C has functors F : C → D as objects and natural transformations as morphisms. Remark 1.5. Any Grothendieck universe models ZFC, in particular Replacement. This ensure that for any two functors F, G : C → D, the class of natural transformation φ : F ⇒ G is a set. This means that Fun(C , D) is locally small, a condition of our definition of a category. Definition 1.6. Given a category C , the arrow category Ar(C ) of C has as objects morphisms f : X0 → X1 in C and as morphisms M :(f : X0 → X1) → (g : Y0 → Y1) the pairs (M0,M1) of morphisms M0 : X0 → Y0 and M1 : X1 → Y1 such that f X0 X1 M0 M1 Y0 g Y1 commutes. Note 1.7. 1. Ar(C ) =∼ Fun([1], C ). 2. Fun(C × D, E ) =∼ Fun(C , Fun(D, E )). 2 2 Equivalences Usually, it is useful to make our notion of sameness between categories weaker than isomorphism. Definition 2.1. A functor F : C → D is an equivalence if there is a functor G : D → C , called the ∼ ∼ quasi-inverse of F , such that F ◦ G = IdC and G ◦ F = IdD . In this case, we say that F and G are equivalent categories. Moreover, we say that a property of C is categorical if it is invariant under categorical equivalence. Example 2.2. Let k be a field. Let the category Matk have natural numbers as objects and morphisms n → p given by p × n matrices over k. Let fdMod denote the category of finite-dimensional vector spaces with linear maps as morphisms. These two categories are equivalent. Indeed, send the natural number n to kn in one direction and the space V to dim V in the other direction. Definition 2.3. A functor F : C → D is essentially surjective if for each object Z of D, there is some object Y of C such that F (Y ) =∼ Z. Theorem 2.4. A functor is an equivalence iff it is full, faithful, and essentially surjective.1 Definition 2.5. A skeleton of C is a full subcategory C 0 ⊂ C such that each element of ob C is isomorphic to exactly one element of ob C 0. An application of Theorem 2.4 yields the following result. Lemma 2.6. Let C 0 be a skeleton of C . Then the inclusion functor C 0 ,→ C is an equivalence. Lemma 2.7. Any two skeleta C 0, C 00 ⊂ C are isomorphic. 0 00 ∼ Proof. Define F : C → C on objects by F (X) = Y where X = Y via a chosen isomorphism hX and on −1 −1 00 0 morphisms f ∈ C (X, Y ) by F (f) = hY ◦ f ◦ (hX ) . To get F , define G : C → C by similarly choosing −1 00 an isomorphism (hX ) for each X ∈ ob C . Remark 2.8. Both Lemma 2.6 and Lemma 2.7 are logically equivalent to the axiom of choice, as is the statement that every category admits a skeleton. 3 Adjunctions Definition 3.1. op 1. Let Z ∈ ob C . Define the contravariant functor YZ : C → Set on objects by Y 7→ C (Y, Z) and on morphisms by sending f : X → Y in C to the map f ∗ : C (Y, Z) → C (X, Z) given by g 7→ gf. We call C (−,Z) := Y Z the set-valued functor represented by Z in C . 2. Let X ∈ ob C . Define the functor Y X : C → Set on objects by Y 7→ C (X, Y ) and on morphisms by sending g : Y → Z to the map g∗ : C (X, Y ) → C (X, Z) given by f 7→ gf. We call C (X, −) := Y X the set-valued functor corepresented by X in C . 1Theorem 3.2.10 (Rognes). 3 A functor of the form C × C 0 → D is called a bifunctor. In particular, define C (−, −): C op × C → Set on objects by (X, X0) → C (X, X0) and on morphisms by sending (f, f 0):(X, X0) → (Y, Y 0) to the map C (f, f 0): C (X, X0) → C (Y, Y 0) given by g 7→ f 0gf. Let C and D be categories and F : C → D and G : D → C be functors. Definition 3.2 (Kan). Consider the set-valued bifunctors D(F (−), −), C (−,G(−)) : C op × D → Set. An adjunction from F to G is a natural isomorphism φ : D(F (−), −) ⇒ C (−,G(−)). If such a φ exists, then we say that (F, G) is an adjoint pair (of functors). Note that φ is natural in the sense that for any map c : X0 → X in C and d : Y → Y 0 in D, the square φX,Y D(F X, Y ) C (X, GY ) ∗ ∗ c d∗ c d∗ D(FX0,Y 0) C (X0, GY 0) φX0,Y 0 commutes in Set. Example 3.3. Let (P, ≤) and (Q, ≤) be preorders. An adjoint pair (F : P → Q, G : Q → P ) is precisely a pair of order-preserving functions such that F x ≤ y ⇐⇒ x ≤ Gy for all x ∈ P and y ∈ Q. In order theory, such a pair is called a Galois connection. Proposition 3.4. Left and right adjoints are both unique up to unique isomorphism. Terminology. We call F the left adjoint to G and G the right adjoint to F . In symbols, F a G. Note 3.5. It is straightforward to check that any adjoint triple F a G a H yields two new adjunctions: GF a GH FG a HG Definition 3.6. Given an adjunction φ : D(F (−), −) ⇒ C (−,G(−)), define the unit morphism ηX = φX,F X (IdFX ) ∈ C (X, GF (X)) and the counit morphism −1 Y = φGY,Y (IdGY ) ∈ D(FG(Y ),Y ). Lemma 3.7. The unit morphisms (ηX )X∈ob C define a natural transformation η : IdC ⇒ GF , and the counit morphisms (Y )Y ∈ob D define a natural transformation : FG ⇒ IdD . 4 Proof. For simplicity, let us just prove that is a natural transformation. We must check that FG(y) FG(Y ) FG(Y 0) Y Y 0 0 Y y Y commutes for any map y : Y → Y 0 in D. By the naturality of φ, we have that −1 y ◦ Y = y ◦ φ (IdGY ) −1 = φ (Gy ◦ IdGY ) −1 = φ (IdGY 0 ◦Gy) −1 = φ (IdGY 0 ) ◦ FG(y) = Y 0 ◦ FG(y), as required.