natural transformation




Just as a functor is a morphism between categories, a natural transformation is a 2-morphism between two functors.

Natural transformations are the 2-morphisms in the 2-category Cat.


Explicit definition

Given categories CC and DD and functors F,G:CD,F,G \colon C \to D, a natural transformation α:FG\alpha \colon F \Rightarrow G between them, denoted

is an assignment to every object xx in CC of a morphism α x:F(x)G(x)\alpha_x:F(x) \to G(x) in DD (called the component of α\alpha at xx) such that for any morphism f:xyf:x \to y in CC, the following diagram commutes in DD:

F(x) F(f) F(y) α x α y G(x) G(f) G(y). \array{ F(x) & \stackrel{F(f)}{\to} & F(y) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(x) & \stackrel{G(f)}{\to} & G(y) } \,.


Natural transformations between functors CDC \to D and DED \to E compose in the obvious way to natural transformations CEC \to E (this is their vertical composition in the 2-category Cat) and functors F:CDF : C \to D with natural transformations between them form the functor category

[C,D]Cat [C,D] \in Cat

The notation alludes to the fact that this makes Cat a closed monoidal category. Since CatCat is in fact a cartesian closed category, another common notation is D CD^C. In fact, if we want CatCat to be cartesian closed, the definition of natural transformation is forced (since an adjoint functor is unique). This is discussed in a section below.

There is also a horizontal composition of natural transformations, which makes Cat a 2-category: the Godement product. See there for details.

In fact, Cat is a 2-category (a CatCat-enriched category) because it is (cartesian) closed: closed monoidal categories are automatically enriched over themselves, via their internal hom.

An alternative but ultimately equivalent way to define a natural transformation α:FG\alpha : F \rightarrow G is as an assignment to every morphism m:xym : x \rightarrow y in CC of a morphism α(m):F(x)G(y)\alpha(m) : F(x) \rightarrow G(y), in such a way as that α(m 0m 1m 2)=G(m 0)α(m 1)F(m 2)\alpha(m_0 m_1 m_2) = G(m_0) \alpha(m_1) F(m_2) for every ternary composition m 0m 1m 2m_0m_1m_2 in CC. The relation of this to the previous definition is that the commutative squares in the previous definition for any morphism ff give the value α(f)\alpha(f), and each α(id x)\alpha(id_x) gives the component α x\alpha_x. Composition of natural transformations can be specified directly in terms of this account as well: specifically, an nn-ary composition α 1...α n\alpha_1 ... \alpha_n of natural transformations is uniquely determined by the property that (α 1...α n)(m 1...m n)=α 1(m 1)...α n(m n)(\alpha_1 ... \alpha_n)(m_1 ... m_n) = \alpha_1(m_1) ... \alpha_n(m_n), for every nn-ary composition m 1...m nm_1 ... m_n in CC.

In terms of the cartesian closed monoidal structure on CatCat

The definition of the functor category [C,D][C,D] with morphisms being natural transformations is precisely the one that makes CatCat a cartesian closed monoidal category.

The category Cat of all categories (regarded for the moment just as an ordinary 1-category) is a cartesian monoidal category: for every two categories CC and DD there is the cartesian product category C×DC \times D, whose objects and morphisms are simply pairs of objects and morphisms in CC and DD: Mor(C×D)=Mor(C)×Mor(D)Mor(C \times D) = Mor(C) \times Mor(D).

It therefore makes sense to ask if there is for each category CCatC \in Cat an internal hom functor [C,]:CatCat[C,-] : Cat \to Cat that would make Cat into a closed monoidal category in that for A,B,CCatA,B,C \in Cat we have natural isomorphisms of sets of functors

Funct(A×C,B)Funct(A,[C,B]). Funct(A \times C , B) \simeq Funct(A, [C,B]) \,.

This is precisely the case for [C,B][C,B] being the functor category with functors CBC \to B as objects and natural transformations, as defined above, as morphisms.

Since CatCat here is cartesian closed, one often uses the exponential notation B C:=[B,C]B^C := [B,C] for the functor category.

To derive from this the definition of natural transformations above, it is sufficient to consider the interval category A:=I:={ab}A := I := \{a \to b\}. For any category EE, a functor IEI \to E is precisely a choice of morphism in EE. This means that we can check what a morphism in the internal hom category [C,B][C,B] is by checking what functors I[C,B]I \to [C,B] are. But by the defining property of [C,B][C,B] as an internal hom, such functors are in natural bijection to functors I×CBI \times C \to B.

Funct(I,[C,B])Funct(I×C,B). Funct(I, [C,B]) \simeq Funct(I \times C, B) \,.

But, as mentioned above, we know what the category I×CI \times C is like: its morphisms are pairs of morphisms in II and CC, subject to the obvious composition law, which says in particular that for f:c 1c 2f : c_1 \to c_2 any morphism in CC we have

(c 1,a)(f,(ab))(c 2,b) =(c 1,a)(f,Id)(c 2,a)(Id,(ab)(c 2,b) =(c 1,a)(Id,(ab))(c 1,b)(f,Id(c 2,b). \begin{aligned} (c_1,a) \stackrel{(f,(a \to b))}{\to} (c_2,b) & = (c_1,a) \stackrel{(f, Id)}{\to} (c_2,a) \stackrel{(Id, (a \to b)}{\to} (c_2, b) \\ &= (c_1,a) \stackrel{(Id, (a\to b))}{\to} (c_1,b) \stackrel{(f,Id}{\to} (c_2, b) \end{aligned} \,.

Here the right side is more conveniently depicted as a commuting square

(c 1,a) (f,Id) (c 2,a) (Id,(ab)) (Id,(ab)) (c 1,b) (f,Id) (c 2,b). \array{ (c_1,a) &\stackrel{(f,Id)}{\to}& (c_2,a) \\ \downarrow^{\mathrlap{(Id,(a \to b))}} && \downarrow^{\mathrlap{(Id, (a \to b))}} \\ (c_1,b) &\stackrel{(f,Id)}{\to}& (c_2,b) } \,.

So a natural transformation between functors CDC \to D is given by the images of such squares in DD. By tracing back the way the hom-isomorphism works, one finds that the image of such a square in DD for a natural transformation α:FG\alpha : F \to G is the naturality square from above:

F(c 1) F(f) F(c 2) α x α y G(c 1) G(f) G(c 2). \array{ F(c_1) & \stackrel{F(f)}{\to} & F(c_2) \\ \alpha_x\downarrow && \downarrow \alpha_y \\ G(c_1) & \stackrel{G(f)}{\to} & G(c_2) } \,.

In terms of double categories

There is a nice way of describing these structures due to Charles Ehresmann. For a category DD let (D, 1, 2)(\square D,\circ_1,\circ_2) be the double category of commutative squares in DD. Then the class of natural transformations of functors CDC \to D can be described as Cat(C,(D, 1))Cat(C,(\square D,\circ_1)). But then 2\circ_2 induces a category structure on this and so we get CAT(C,D)CAT(C,D).

An advantage of this approach is that it applies to the case of topological categories and groupoids (working in a convenient category of spaces).

An analogous approach works for strict cubical ω\omega-categories with connections, using the good properties of cubes, so leading to a monoidal closed structure for these objects. This yields by an equivalence of categories a monoidal closed structure on strict globular omega-categories, where the tensor product is the Crans-Gray tensor product.


For functors between higher categories, see lax natural transformation etc.

A transformation which is natural only relative to isomorphisms may be called a canonical transformation.

For functors with more complicated shapes than CDC \rightrightarrows D, see extranatural transformation and dinatural transformation.

Last revised on April 3, 2019 at 03:15:34. See the history of this page for a list of all contributions to it.