1 Natural Transformations

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(FX,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.

Moreover, one can verify that the unit and counit of φ satisfy the triangle identities,

FX ◦ F ηX = 1FX (M1)

GY ◦ ηGY = 1GY , (M2) for any X ∈ ob C and Y ∈ ob D. Conversely, suppose that F and G come equipped with two natural transformations

η : IdC ⇒ GF

 : FG ⇒ IdD satisfying the triangle identities. Then we get an adjunction φ from F to G with component

φX,Y : D(FX,Y ) → C (X, GY ), f 7→ Gf ◦ ηX .

Indeed, define ψX,Y : C (X, GY ) → D(FX,Y ) by g 7→ Y ◦ F g. We have that

ψX,Y (φX,Y (f)) = ψX,Y (Gf ◦ ηX )

= Y ◦ F (Gf ◦ ηX )

= Y ◦ F (Gf) ◦ F ηX

= f ◦ FX ◦ F ηX (naturality of )

= f. ((M1))

Likewise, we have that φX,Y (ψX,Y (g)) = g. Hence φX,Y is a natural isomorphism in both X and Y with inverse ψX,Y . F Even so, C  D need not be an equivalence of categories, as η and  may not be isomorphisms. Further, G L a given equivalence C  D of categories need not be an adjunction, as its associated natural transformations R

0 η : IdC ⇒ RL 0  : LR ⇒ IdD

5 may not satisfy the triangle inequalities. Nevertheless, (L, R) is an adjoint pair with unit η0 and counit another natural transformation defined in terms of η0 and 0. By symmetry, (R,L) is also an adjoint pair.

Example 3.8 (Monad). Let (C , ⊗, 1) be a monoidal category. A monoid in C is an object M equipped with a multiplication map µ : M ⊗ M → M and a unit map η : 1 → M that satisfy certain coherence properties expressing that µ is associative and that η is a two-sided identity. Given two monoids (M, µ, η) and (M 0, µ0, η0) in C , a map f : M → M 0 in C is a morphism of monoids if it satisfies

f ◦ µ = µ0 ◦ (f ⊗ f) f ◦ η = η0.

A comonoid N in C is a monoid in C op, equipped with a comultiplication map δ : N → N 2 and a counit map  : N → 1.

For example, a monoid in the monoidal category (End(C ), ◦, IdC ) of endofunctors of C is called a monad on C . A comonoid in End(C ) is called a comonad on C . Explicitly, a monad on C consists of an endofunctor T : C → C together with two natural transformations 2 η : IdC → Y and µ : T → T such that the following diagrams commute:

T µ η T η T 3 T 2 T T T 2 T

µT µ µ .

2 T µ T T

These are precisely the associativity and unit laws, respectively. Now, let (F : C → D,G : D → C ) be an

adjoint pair with unit η : IdC : G ◦ F and counit  : F ◦ G → IdD . We then have a natural transformation (G ◦ F )2 → G ◦ F given componentwise by

G (FX ): GF GF X → GF X

One can check that (G ◦ F, η, GF ) is a monad on C . Dually, a comonad R : C → C on C satisfies the relations

δR ◦ δ = Rδ ◦ δ

R ◦ δ = IdR = R ◦ δ.

Moreover, any adjoint pair (F : C → D,G : D → C ) with unit η and counit  induces a comonad (G, , δ) on D where

G ≡ F ◦ G : D → D 2 δ ≡ F ηG : G → G .

Theorem 3.9. The category of monoids in C is equivalent to the category of C -enriched categories with one object.

Example 3.10.

(1) The forgetful functor U : Grp → Set has a left adjoint F : Set → Grp sending a set to the free group generated by A.

6 (2) Let R be a ring. The forgetful functor U : R−Mod → Set has a left adjoint R(−) sending a set S to L s∈S R, the free R-module generated by S.

The forgetful functor has no right adjoint in either Example 3.10(1) or Example 3.10(2). It does, however, have one in the following setting.

Example 3.11. The forgetful functor U : Top → Set has a left adjoint that sends a set to the same set equipped with the discrete topology. It also has a right adjoint that sends a set to the same set equipped with the indiscrete topology.

Definition 3.12. A subcategory C ⊂ D is reflective if the inclusion functor has a left adjoint and is coreflective if the inclusion functor has a right adjoint.

Example 3.13.

1. The full subcategory Ab ⊂ Grp is reflexive as the inclusion functor is right adjoint to (−)ab.

2. Let AbT ⊂ Ab denote the subcategory of torsion groups. This is coreflective as the inclusion functor is right adjoint to the functor sending an abelian group to its torsion subgroup.

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