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


(natural transformations)
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:


(composition of natural transformations) 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

(1)[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, cf. exponential objects.)

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.


(categories of presheaves)
The functor categories (1) are also called categories of presheaves, in particular if they are of the form

PSh(C)[C op,Set], PSh(C) \,\coloneqq\, [C^{op}, Set] \,,

hence if they are categories whose

Similarly, functor categories of the form

CPSh(C)[C,Set] CPSh(C) \,\coloneqq\, [C, Set]

are also called categories of copresheaves.

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.



(unit of double-dualization)
For any ground field 𝕂\mathbb{K}, the canonical linear map from a vector space VV to its double dual (V *) *\big(V^{\ast}\big)^\ast


is a natural transformation

Id(() *) * Id \longrightarrow \big((-)^{\ast}\big)^\ast

from the identity functor IdId on 𝕂 Vect \mathbb{K} Vect to the composition (() *) *\big((-)^{\ast}\big)^\ast of the linear dual-endofunctor with itself.

This example was the motivating example of Eilenberg & MacLane 1945 (right on the first pages) for introducing the notion of natural transformation (and with it category theory) in the first place.

The conceptual subtlety that these authors sought to resolve here is that for any finite-dimensional vector space there exists also an isomorphism from VV to its single-dual vector space:

V𝕂Vect fdimVV *. V \in \mathbb{K} Vect^{fdim} \;\;\;\;\;\; \vdash \;\;\;\;\;\; \exists \; V \xrightarrow{\sim} V^\ast \,.

But these linear maps are conceptually different from (2) in that they are not natural in the technical sense that they do not form the components of a natural transformation between the evident functors. Instead they involve an arbitrary choice equivalent to that of an (possibly indefinite) inner product on VV, which is not preserved by general linear isomorphisms (but just by the corresponding isometries).


The determinant is a natural transformation det:GL n() ×det\colon GL_n\rightarrow (-)^\times from the general linear group to the group of units of a ring, which are both functors from Ring to Grp.


The Frobenius homomorphism is a natural transformation Frob p:Id Ring pId Ring pFrob_p\colon Id_{Ring_p}\Rightarrow Id_{Ring_p} from the identity functor on the full subcategory of Ring containing all rings with characteristic pp to itself.


The Hurewicz homomorphism is a natural transformation h n:π nH n(;)h_n\colon\pi_n\Rightarrow H_n(-;\mathbb{Z}) from the homotopy group to singular homology, which are both functors from Top to Grp.


The inversion GG op,gg 1G \rightarrow G^{op},g\mapsto g^{-1} for every group GG yields a natural transformation Id Grp() op\Id_{Grp}\Rightarrow (-)^{op} from the identity functor on Grp to the opposite group-assigning functor.


The coprojection GG ab,g[g]G\rightarrow G^{ab},g\mapsto[g] for every group GG yields a natural transformation Id Grp ab\Id_{Grp}\Rightarrow -^{ab} from the identity functor on GrpGrp to the abelianization functor.


(homomorphisms of diagrams)
By Remark , every category identified with a category of presheaves or copresheaves has its morphisms identified with natural transformations.

For instance, the category of directed graphs (digraphs) may be identified with the category of copresheaves on the diagram shape {VE}\big\{ V \rightrightarrows E\big\}, and under this identification the natural transformations between functors {VE}Set\big\{ V \rightrightarrows E \big\} \longrightarrow Set are identified with digraph homomorphisms.


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.


The notion of natural transformations between functors is due to

where it served as the motivation for the definition of categories and functors in the first place.

Textbook accounts:

See also category theory - references.

Last revised on June 13, 2024 at 09:11:25. See the history of this page for a list of all contributions to it.