f //C0
γ0
T C T f //T C0
commutes. Prove that if(C, γ)is aterminal coalgebra, that is a terminal object in the category of coalgebras, then the mapγ:C→T Cis an isomorphism.
1.7. The 2-category of categories
A number of important facts about natural transformations are proven by diagram chasing. In this section, we define “vertical” and “horizontal” composition operations for natural transformations. The upshot is that categories, functors, and natural transformations assemble into a 2-dimensional categorical structure called a2-category, a definition that is stated at the conclusion.
In French, a natural transformation is called amorphisme de foncteurs. Indeed, for any fixed pair of categoriesCandD, there is afunctor categoryDCwhose objects are functors C→Dand whose morphisms are natural transformations. Given a functorF:C→D, its identity natural transformation1F:F⇒Fis the natural transformation whose components (1F)cB1Fcare identities. The following lemma defines the composition of morphisms in DC.
Lemma 1.7.1 (vertical composition). Suppose α: F ⇒ G and β:G ⇒ H are natural transformations between parallel functors F,G,H: C → D. Then there is a natural transformationβ·α: F⇒Hwhose components
(β·α)cBβc·αc
are defined to be the composites of the components ofαandβ.
Proof. Naturality ofαandβimplies that for any f:c→c0in the domain category, each square, and thus also the composite rectangle, commutes:
Fc
F f
αc //Gc
G f
βc //Hc
H f
Fc0 α
c0 //Gc0
βc0 //Hc0
Corollary 1.7.2. For any pair of categoriesCandD, the functors fromCtoDand natural transformations between them define a categoryDC.
Proof. It remains only to verify that the composition operation defined by Lemma 1.7.1 is associative and unital. It suffices to verify these properties componentwise, and they follow immediately from the associativity and unitality of composition inD. Remark 1.7.3 (sizes of functor categories). Care should be taken with size when discussing functor categories. IfCandDare small, thenDCis again a small category, but ifCand Dare locally small, thenDCneed not be. This is only guaranteed ifDis locally small and Cis small; see Exercise 1.7.i. In summary, the formation of functor categories defines a
1.7. THE 2-CATEGORY OF CATEGORIES 45
bifunctorCatop×Cat→CatorCatop×CAT→CAT, but the category of functors between two non-locally small categories may be even larger than these categories are.
The composition operation defined in Lemma 1.7.1 is calledvertical composition. Drawing the parallel functors horizontally, a composable pair of natural transformations in the categoryDCfits into apasting diagram
C
As the terminology suggests, there is also ahorizontal compositionoperation
C
Lemma 1.7.4 (horizontal composition). Given a pair of natural transformations
C
there is a natural transformationβ∗α: HF ⇒KGwhose component atc∈Cis defined as the composite of the following commutative square
(1.7.5)
Proof. The square (1.7.5) commutes by naturality ofβ: H⇒ Kapplied to the mor-phismαc:Fc→GcinD. To prove that the components(β∗α)c: HFc→KGcso-defined are natural, we must show thatKG f ·(β∗α)c=(β∗α)c0·HF f for any f:c→c0inC. This relation holds on account of the commutative rectangle
HFc
The right-hand square commutes by naturality ofβ. The left-hand square commutes by naturality ofαand Lemma 1.6.5, which states that functors, in this case the functor H,
preserve commutative diagrams.42
Remark 1.7.6. The natural transformations
Hα: HF⇒HG, Kα:KF ⇒KG, βF:HF⇒KF, and βG:HG⇒KG appearing in Lemma 1.7.4 are defined bywhiskeringthe natural transformationsαand β with the functors H and K or F and G, respectively. A precise definition of this construction is given in Exercise 1.7.ii. The terminology is on account of the following graphical depiction of the whiskered composite
LβF:LHF⇒LKF C F //D
of the natural transformationβ:H⇒Kwith the functorsFandL. Exercise 1.7.iii explains the particular interest in the case where eitherLorFis an identity.
Importantly, vertical and horizontal composition can be performed in either order, satisfying the rule ofmiddle four interchange:
Lemma 1.7.7 (middle four interchange). Given functors and natural transformations
C
the natural transformationJF⇒LHdefined by first composing vertically and then compos-ing horizontally equals the natural transformation defined by first composcompos-ing horizontally and then composing vertically:
Lemmas 1.7.1, 1.7.4, and 1.7.7 prove that categories, functors, and natural transfor-mations assemble into a2-category. Aside from this example, we will not meet any other 2-categories in this text. Nonetheless, the following definition is useful as an axiomatization of the composition operations for natural transformations that are available. A succinct introduction to 2-categories andpasting diagrams, which are used to display composite natural transformations, can be found in [KS74].
Definition 1.7.8. A2-categoryis comprised of:
42Naturality ofβ∗αcould also be deduced from a second commutative rectangle that defines the component (β∗α)cas the top-right composite of (1.7.5). The point is that the squares (1.7.5) and the morphisms obtained by applying the four functorsHF,HG,KF, andKGto a morphism inCdefine a commutative cube.
1.7. THE 2-CATEGORY OF CATEGORIES 47
• objects, for example the categoriesC,
• 1-morphisms between pairs of objects, for example, the functorsC−→F D, and
• 2-morphisms between parallel pairs of 1-morphisms, for example, the natural transfor-mations C
F %%
G
99
⇓α D
so that:
• The objects and 1-morphisms form a category, with identities1C:C→C.
• For each fixed pair of objectsCandD, the 1-morphismsF:C→Dand 2-morphisms between such form a category under an operation called vertical composition, as described in Lemma 1.7.1, with identities C
F %%
F
99
⇓1F D.
• There is also a category whose objects are the objects in which a morphism fromC toD is a 2-cell C
F %%
G
99
⇓α D under an operation called horizontal composition, with
identities C 1C
%%
1C
99⇓11C C. The source and target 1-morphisms of a horizontal composition must have the form described in Lemma 1.7.4.
• The horizontal composite1H∗1F of identities for vertical composition must be the identity1HFfor for the composite 1-morphisms.
• The law of middle four interchange described in Lemma 1.7.7 holds.
The reader who has taken the categorical philosophy to heart might ask: What is a morphism between 2-categories?2-functorswill make an appearance in §4.4.
Exercises.
Exercise 1.7.i. Prove that ifCis small andDis locally small, thenDCis locally small by defining a monomorphism from the collection of natural transformations between a fixed pair of functors F,G: C ⇒ D into a set. (Hint: Think about the function that sends a natural transformation to its collection of components.)
Exercise 1.7.ii. Given a natural transformation β: H ⇒ K and functors F and L as displayed in
C F //D H
K
EE⇓β E L //F
define a natural transformation LβF: LHF ⇒ LKF by (LβF)c = LβFc. This is the whiskered compositeofβwithLandF. Prove thatLβFis natural.
Exercise 1.7.iii. Redefine the horizontal composition of natural transformations intro-duced in Lemma 1.7.4 using vertical composition and whiskering.
Exercise 1.7.iv. Prove Lemma 1.7.7.
Exercise 1.7.v. Show that for any categoryC, the collection of natural endomorphisms of the identity functor1Cdefines a commutative monoid, called thecenter of the category. The proof of Proposition 1.4.4 demonstrates that the center ofAbfgis the multiplicative monoid(Z,×,1).
Exercise 1.7.vi. Suppose the functors and natural isomorphisms C F //D
oo G η: 1CGF :FG1D
D F
0 //E
G0
oo η0: 1DG0F0 0:F0G01E
define equivalences of categoriesC'DandD'E. Prove (again) that there is a composite equivalence of categoriesC'Eby defining composite natural isomorphisms1CGG0F0F andF0FGG01E.
Exercise 1.7.vii. Prove that a bifunctorF:C×D→Edetermines and is uniquely deter-mined by:
(i) A functorF(c,−) :D→Efor eachc∈C.
(ii) A natural transformation F(f,−) : F(c,−) ⇒ F(c0,−) for each f: c → c0 in C, defined functorially inC.
In other words, prove that there is a bijection between functorsC×D → Eand functors C→ ED. By symmetry of the product of categories, these classes of functors are also in bijection with functorsD→EC.
CHAPTER 2