Natural Transformations and Functor Categories

Categories have maps between them, the functors. Functors in turn have maps between them, and this second-level notion is the one category theory was invented for: Eilenberg and Mac Lane introduced categories and functors in order to say precisely what a natural construction is. A natural transformation compares two parallel functors $F, G : A \to B$ by giving, for every object of $A$ , an arrow from its $F$ -image to its $G$ -image, uniformly in a sense the definition makes exact.

The definition

To see what the compatibility condition should be, start where no condition is needed. Let $A$ be the discrete category with objects $0, 1, 2, \dots$ ; functors $F, G : A \to B$ are just sequences $(F_{0}, F_{1}, \dots)$ and $(G_{0}, G_{1}, \dots)$ of objects of $B$ , and the reasonable notion of map from $F$ to $G$ is a sequence of arrows $α_{i} : F_{i} \to G_{i}$ . In general $A$ has arrows too, and the components must respect them.¹

The naturality square for the component of a transformation

t

at an arrow

f : A \to B

: pushing forward along

F (f)

then crossing over equals crossing over then pushing along

G (f)

As with categories and functors, the definition is arranged so that each arrow $f : A \to A^{'}$ of $A$ yields exactly one arrow $F (A) \to G (A^{'})$ of $B$ — the common diagonal of the square. We write $α : F \Rightarrow G$ when the double arrow is helpful for distinguishing levels.

The square rules out case-by-case definitions. A family of maps defined by one formula for some objects and another formula for others will generically fail the square for arrows crossing between the cases. Naturality is the formal counterpart of defined the same way for every object.

Worked examples

The functor category

Natural transformations compose. Given $α : F \Rightarrow G$ and $β : G \Rightarrow H$ (with $F, G, H : A \to B$ ), define $(β \circ α)_{A} = β_{A} \circ α_{A}$ ; the two-storey naturality squares stack into one tall rectangle, which commutes because each storey does. Each functor $F$ has an identity transformation $1_{F}$ with components $1_{F (A)}$ .

Vertical composition stacks naturality squares; the outer rectangle commutes because both inner squares do, giving components

u_{A}

then

v_{A}

The exponent notation is justified on small examples. If $2$ is the discrete two-object category, $[2, B] ≅ B \times B$ , a pair of objects and a pair of maps, matching $B^{2}$ . If $G$ is a monoid, $[G, Set]$ is the category of left $G$ -sets and equivariant maps, and $[G^{op}, Set]$ the right $G$ -sets. If $A$ and $B$ are ordered sets, $[A, B]$ is again an ordered set: the monotone maps, compared pointwise ( $f \leq g$ iff $f (a) \leq g (a)$ for all $a$ ; at most one transformation exists because all diagrams in a thin category commute).

Natural isomorphism

Since $[A, B]$ is a category, it comes with a notion of isomorphism automatically.

One direction is immediate. For the other, the inverses $α_{A}^{- 1}$ must be shown to form a natural family; rearranging the naturality square of $α$ around $f$ does it. The double dual above is the flagship instance: the canonical $α_{V}$ are isomorphisms, so $1_{FDVect} ≅ (-)^{**}$ as functors.

A natural isomorphism is a bridge of parallel component isomorphisms

t_{A_{i}} : F (A_{i}) \to G (A_{i})

, each invertible, between the two images of the source category; every square between planks commutes.

$F (A) ≅ G (A)$ for every $A$ separately is strictly weaker than $F ≅ G$ . The pointwise statement supplies isomorphisms; naturality demands they be chosen coherently, and sometimes no coherent choice exists (the single dual $V^{*}$ again). When the source category is discrete the two notions collapse (there are no squares to satisfy), which is why the distinction is invisible until functors enter.

Natural isomorphism also repairs the too-strict notion of sameness for categories. Isomorphism of categories ( $GF = 1$ , $F G = 1$ on the nose) almost never occurs; the correct notion, equivalence, asks only $G \circ F ≅ 1_{A}$ and $F \circ G ≅ 1_{B}$ , with natural isomorphisms in place of equalities.⁸ A functor is an equivalence precisely when it is full, faithful, and essentially surjective — the classification set up in the functors lesson. Equivalence is the sense in which the category of finite sets and the category of sets ${1, \dots, n}$ are the same, and in which one-object categories with invertible arrows are groups.

Horizontal composition and the Godement calculus

Vertical composition stacks transformations between functors sharing source and target. There is a second, orthogonal composition when the functors themselves compose. Given

A G ⇉ F B K ⇉ H C, α : F \Rightarrow G, β : H \Rightarrow K,

two hybrid whiskerings are available first:⁹

$β F : H \circ F \Rightarrow K \circ F$ with components $(β F)_{A} = β_{F (A)}$ — sample $β$ at image objects; naturality is $β$ 's own naturality at the arrows $F (f)$ .
$H α : H \circ F \Rightarrow H \circ G$ with components $(H α)_{A} = H (α_{A})$ — apply $H$ to the components; naturality is $H$ preserving $α$ 's squares.

The two constructions are semantically different (one indexes into $β$ , the other transports $α$ ) yet formally interchangeable, which is why the notation treats them symmetrically. The horizontal composite $β * α : H \circ F \Rightarrow K \circ G$ is defined by either route around the square below, and naturality of $β$ makes the routes agree:

(β * α)_{A} = K (α_{A}) \circ β_{F (A)} = β_{G (A)} \circ H (α_{A}) .

Horizontal composition of

a = α

and

b = β

: two transformations across consecutive functor pairs compose to one across the composites; the square commutes by naturality, and its diagonal is the component of the composite at

A

Horizontal composition is associative, and it meshes with vertical composition through the rule that organizes every calculation involving both:

Composing vertically then horizontally equals composing horizontally then vertically. Godement codified this together with four bookkeeping rules for mixing functors and transformations (e.g. $(H \circ G) α = H (G α)$ and $γ (F \circ E) = (γ F) E$ ); all five reduce to the interchange law plus whiskering-as-horizontal-composition with identities, $β F = β * 1_{F}$ and $H α = 1_{H} * α$ . The interchange law is the defining coherence of a 2-category (objects, arrows, and arrows between arrows), of which $CAT$ is the founding example. We will not develop 2-categories, but the law itself returns in the Godement-style computations with monads, where $μ$ and $η$ are natural transformations composed both ways at once.

Combining categories, functors, and natural transformations

Category, functor, and natural transformation together support the constructions of later modules:

Functor categories. $[A^{op}, Set]$ , the category of presheaves, is where representable functors live; the Yoneda lemma computes its hom-sets out of representables.
Natural isomorphism as the right equality. Statements like $D (F A, B) ≅ C (A, GB)$ naturally in $A$ and $B$ — the definition of an adjunction — only make sense with natural transformations in hand.
Limits in functor categories are computed componentwise, one naturality square at a time, in the limits module.

Leinster, Basic Category Theory, §1.3, Definition 1.3.1 and the discrete-category motivation preceding it — components, the naturality square, and the exactly-one-arrow $F (A) \to G (A^{'})$ bookkeeping of Remarks 1.3.2. ↩
Leinster, §1.3, Example 1.3.5 — the determinant as a natural transformation $M_{n} \Rightarrow U$ between functors $CRing \to Mon$ ; naturality as defined in the same way for all rings. ↩
Leinster, §1.3, Example 1.3.14 — the canonical evaluation map $V \to V^{**}$ as a natural isomorphism from the identity to the double-dual functor on $FDVect_{k}$ . ↩
Leinster, §1.3, Example 1.3.4 — natural transformations between functors $G \to Set$ recover the $G$ -equivariant maps. ↩
Barr & Wells, Category Theory for Computing Science, §4.3.8 — flatten as a natural transformation from the list-of-lists functor to the list functor, and reduce from $F \circ U$ to the identity on $Mon$ . ↩
Leinster, §1.3, Construction 1.3.6 and Examples 1.3.7–1.3.9 — vertical composition, identity transformations, the functor category $[A, B]$ , and the computations of $[2, B]$ , $[G, Set]$ , and $[A, B]$ for ordered sets. ↩
Leinster, §1.3, Definition 1.3.10, Lemma 1.3.11, and Example 1.3.13 — natural isomorphism as isomorphism in the functor category, the componentwise criterion, and the warning that pointwise isomorphism does not imply natural isomorphism. ↩
Barr & Wells, §4.3.4 — a functor is an equivalence with pseudo-inverse $G$ iff $G \circ F$ and $F \circ G$ are naturally isomorphic to the identities. ↩
Barr & Wells, §4.4 — the Godement calculus: whiskerings $β F$ and $H α$ (Definitions 4.4.2–4.4.3), horizontal composition and its associativity (4.4.4–4.4.6), and the five rules with the interchange law G-1. ↩