nLab 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:


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.

In terms of morphismwise components

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 G(m 1)α(m 0)=α(m 1)F(m 0)G(m_1)\alpha(m_0) = \alpha(m_1)F(m_0) for every binary composition m 1m 0m_1 m_0 in CC (or equivalently α(m 2m 1m 0)=G(m 2)α(m 1)F(m 0)\alpha(m_2 m_1 m_0) = G(m_2) \alpha(m_1) F(m_0) for every ternary composition m 2m 1m 0m_2m_1m_0 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)=G(f)α x=α yF(f)\alpha(f) = G(f) \circ \alpha_x = \alpha_y \circ F(f), and the identity morphisms for any object xx give the component α x=α(id x)\alpha_x = \alpha(id_x).

Vertical 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.

Horizontal composition is even easier, as the horizontal composite of α 1,...,α n\alpha_1, ..., \alpha_n is just α 1...α n\alpha_1 ... \alpha_n.

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 C B:=[B,C]C^B := [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.


The following properties come from the HoTT book.

It follows that the type of natural transformations from FF to GG is a set.

Composites with functors

For functors F:ABF : A \to B and G,H:BCG,H : B \to C and a natural transformation γ:GH\gamma : G \to H, the composite (γF):GFHF(\gamma F) : G F \to H F is given by

  • For each a:Aa:A, the component γ Fa\gamma_{F a}.

Naturality is easy to check. Similarly, for γ\gamma as above and K:CDK : C \to D, the composite (Kγ):KGKH(K \gamma): K G \to K H is given by

  • For each b:Bb: B, the component K(γ b)K(\gamma_b).

Lemma 9.2.8 in the HoTT book

For functors F,G:ABF,G: A\to B and H,K:BCH,K: B \to C and natural transformations γ:FG\gamma : F \to G and δ:HK\delta : H \to K, we have

(δG)(Hγ)=(Kγ)(δF).(\delta G)(H \gamma) = (K \gamma) (\delta F).

Proof. It suffices to check componentwise: at a:Aa:A we have

((δG)(Hγ)) a (δG) a(Hγ) a δ GaH(γ a) =K(γ a)δ Fa(bynaturalityofδ) (Kγ) a(δF) a ((Kγ)(δF)) a. \begin{aligned} ((\delta G)(H \gamma))_a &\equiv (\delta G)_a (H \gamma)_a \\ &\equiv \delta_{G a} \circ H(\gamma_a)\\ &= K(\gamma_a) \circ \delta_{F a} \qquad (by\ naturality\ of\ \delta) \\ &\equiv (K \gamma)_a \circ (\delta F)_a\\ &\equiv ((K \gamma)(\delta F))_a.\ \square \end{aligned}


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.


Textbook accounts:

See also category theory - references.

Last revised on September 17, 2023 at 06:43:00. See the history of this page for a list of all contributions to it.