We collect here, mostly without proof, some of the basic combinatorial prop-erties of functors and natural transformations. These rules were first codi-fied by Godement [Godement, 1958]. The notation given in Definitions 4.4.2 and 4.4.3 is used throughout the book, but the remainder of this section is used only in Section 4.8. Verifying (some of) the Godement properties is an excellent way to familiarize yourself with natural transformations.
4.4.1 LetF :A −→Band G:B−→C be functors. There is a composite functorG◦F :A −→C defined in the usual way by G◦F(A) =G(F(A)).
Similarly, letH, K and L be functors fromA −→Band α:H −→K and β:K−→Lbe natural transformations. Recall that this means that for each objectAofA,αA:HA−→KAandβA:KA−→LA. Then as in 4.2.11, we defineβ◦α:H −→Lby
Things get more interesting when we mix functors and natural transfor-mations. For example, suppose we have three categoriesA, BandC, four functors, two of them,F, G :A −→B and the other two H, K: B−→C, and two natural transformationsα:F −→G and β :H −→K. We picture this situation as follows:
A F RB
⇓α H RC
4.4.2 Definition The natural transformationβF :H ◦F −→K◦F is de-fined by the formula (βF)A=β(F A) for an object AofA.
The notationβ(F A) means the component of the natural transformation β at the objectF A. This is indeed an arrow fromH(F(A))−→K(F(A)) as required. To show that βF is natural requires showing that for an arrow f :A−→A0 ofA, the diagram
H(F(A0)) -K(F(A0)) βF A0
H(F(A)) βF A-K(F(A))
commutes, but this is just the naturality diagram ofβ applied to the arrow F(f) :F(A)−→F(A0).
4.4.3 Definition The natural transformationHα:H ◦F−→H ◦Gis de-fined by letting (Hα)A=H(αA) for an objectAofA, that is the value of H applied to the arrowαA.
To see thatHαthus defined is natural requires showing that
H(F(A0)) -H(G(A0)) H(αA0)
commutes. This diagram is obtained by applying the functorH to the nat-urality diagram of α. Since functors preserve commutative diagrams, the result follows.
Note that the proofs of naturality forβF and forHαare quite different.
For example, the second requires thatH be a functor, while the first works ifF is merely an object function.
The definitions of βF and Hα are quite different, in fact. The first is the natural transformation whose value at an objectAofA is the component of β on the object F A while the value of the second is the result of applying the functorH to the componentαA(which is an arrow ofB). Nevertheless, we use similar notations. The reason for this is that their formal properties are indistinguishable. In fact, even categorists quite commonly (though not universally) distinguish them by writingβFbutHα. That notation emphasizes the fact that they are semantically different. The notation used here is chosen to emphasize the fact that they are syntactically indistinguishable. More precisely, the left/right mirror image of each of Godement’s rules given below is again a Godement rule.
In a great deal of mathematical reasoning, one forgets the semantics of the situation except at the beginning and the end of the process, relying on the syntactic rules in the intermediate stages. This is especially true in the kind of
‘diagram chasing’ arguments so common in category theory. For that reason, the notation we have adopted emphasizes the syntactic similarity of the two constructions, rather than the semantic difference.
In Exercise 2, we give another, more sophisticated definition of βF andHα which shows that they can be thought of assemantically parallel, as well.
4.4 Godement calculus 119 4.4.4 There is a second way of composing natural transformations. The naturality of β in Diagram (4.29) implies that for any objectA of A, the diagram
(H◦F)A (Hα)A-(H ◦G)A
commutes. We defineβ∗α:H ◦F −→K◦Gby requiring that its component atAbe (Kα)A◦(βF)A, which of course is the same as (βG)A◦(Hα)A.
4.4.5 Proposition β∗αis a natural transformation.
Proof. We have that β∗α=Kα ◦ βF by definition of β∗α and Defini-tion 4.2.11. It is therefore a natural transformaDefini-tion by ProposiDefini-tion 4.2.12.
We usually callβ ◦αthevertical compositeandβ∗αthehorizontal composite. (Warning: Some authors use ◦ for the horizontal composite.) One must keep careful track of the difference between them. Fortunately, the notations do not often clash, since usually only one makes sense.
There is one case in which the two notations can clash. If A =B= C and G=H, then β ◦ α: F −→K, while β∗α:G◦ F −→K ◦ G. This clash is exacerbated by the habit among many categorists of omitting the composition circle and∗, except for emphasis. We will often omit the∗, but not the circle. On the other hand, no confusion can possibly arise from the overloading of the circle notation to include composition of arrows, functors and natural transformations since their domains uniquely define what kind of composition is involved.
4.4.6 Proposition Horizontal composition of natural transformations is associative.
Proof. In the situation,
A F RB G µ
⇓α H RC
⇓β L RD
we have that
γ∗(β∗α) =γKG◦L(βG◦Hα) =γKG◦LβG◦LHα
becauseLis a functor, while
(γ∗β)∗α= (γK ◦Lβ)G◦LHα=γKG◦LβG◦LHα by Definition 4.2.11.
4.4.7 Godement’s five rules There are thus several kinds of composites.
There is a composite of functors, vertical and horizontal composite of natural transformations and the composite of a functor and a natural transformation in either order (although the latter is in fact the horizontal composite of a natural transformation and the identity natural transformation of a functor, a fact we leave to an exercise). The possibilities are sufficiently numerous that it is worth the effort to codify the rules.
LetA,B,C,DandE be categories;E:A −→B,F1,F2andF3:B−→
C,G1,G2andG3:C −→D, andH :D−→E be functors; andα:F1−→F2, β :F2−→F3, γ :G1 −→G2, and δ:G2−→G3 be natural transformations.
This situation is summarized by the following diagram:
A E -B
-C F2 U
⇓β G2 -UD G1
⇓δ H -E (4.32)
G–1 (δ◦γ)(β◦α) = (δβ)◦(γα).
G–2 (H ◦G1)α=H(G1α).
G–3 γ(F1◦E) = (γF1)E.
G–4 G1(β◦α)E= (G1βE)◦(G1αE).
G–5 γα= (γF2)◦(G1α) = (G2α)◦(γF1).
The expression G1(β ◦ α)E in G–4 is not ambiguous because of Exer-cise 1. G–1 is called theInterchange Law. It is the basis for the definition of 2-category in Section 4.8.
1. Show that, using the notation of the Godement rules, (G1α)E=G1(αE).
2. Show that in the Diagram (4.28), the composites βF and Hα are the horizontal compositesβ∗idF and idH∗αrespectively.
3. a. Show (using Exercise 2) that Godement’s fifth rule is an instance of the first.
b.Show that Godement’s fourth rule follows from the first and the asso-ciativity of horizontal composition.
4.5 Yoneda Lemma 121