Natural Transformations

Summary

A natural transformation is a coherent family of morphisms between two functors, one component per object. The naturality condition says these components commute with all morphisms in the domain category. Natural isomorphisms are natural transformations all of whose components are isomorphisms.

Overview

If functors are maps between categories, natural transformations are maps between functors. This gives the three-tier structure of category theory: objects, morphisms, functors, natural transformations. The naturality condition is the categorical coherence condition: it says the transformation “doesn’t make arbitrary choices” as you move through the category.

Main Content

Definition 1.3.1: Natural Transformation

Let $F,G:\mathcal{A}\to \mathcal{B}$ be functors. A natural transformation $\alpha :F\Rightarrow G$ consists of a family of maps $({\alpha }_A:FA\to GA{)}_{A\in \mathcal{A}}$ in $\mathcal{B}$ such that for every $f:A\to A^{\prime }$ in $\mathcal{A}$, the following square commutes:

$$\begin{array}{ccc}FA& \stackrel{{\alpha }_A}{\to }& GA\\ Ff\downarrow & & \downarrow Gf\\ F{A}^{\prime }& \stackrel{{\alpha }_{A^{\prime }}}{\to }& G{A}^{\prime }\end{array}$$

i.e., ${\alpha }_{A^{\prime }}\circ Ff=Gf\circ {\alpha }_A$.

The maps ${\alpha }_A$ are called the components of $\alpha$. We write $\alpha :F\Rightarrow G:\mathcal{A}\to \mathcal{B}$.

Definition: Natural Isomorphism

A natural transformation $\alpha :F\Rightarrow G$ is a natural isomorphism if every component ${\alpha }_A$ is an isomorphism in $\mathcal{B}$. We write $F\cong G$.

Core Examples

Example: Double Dual for Vector Spaces (BCT, Ch. 1.3)

Let $F=1_{{\mathbf{Vect}}_k}$ (identity functor) and $G=(-{)}^{\ast \ast }$ (double dual). There is a natural transformation $\alpha :F\Rightarrow G$ with components ${\alpha }_V:V\to V^{\ast \ast }$ given by $v\mapsto (\varphi \mapsto \varphi (v))$.

Naturality: For any linear $f:V\to W$, the square ${\alpha }_W\circ f=f^{\ast \ast }\circ {\alpha }_V$ commutes. This is “natural” in $V$: no choice of basis is needed.

For finite-dimensional $V$, ${\alpha }_V$ is an isomorphism, giving a natural isomorphism $1\cong (-{)}^{\ast \ast }$ on ${\mathbf{Vect}}_k^{\mathrm{fd}}$.

Example: Determinant (BCT, Ch. 1.3)

The determinant $\det :G{L}_n(-)\Rightarrow (-{)}^{\times }$ is a natural transformation from the functor $G{L}_n:\mathbf{CRing}\to \mathbf{Grp}$ to the units functor $(-{)}^{\times }:\mathbf{CRing}\to \mathbf{Grp}$. Naturality says: for any ring homomorphism $\varphi :R\to S$, $\det (\varphi (M))=\varphi (\det (M))$.

Vertical and Horizontal Composition

Definition: Vertical Composition

If $\alpha :F\Rightarrow G$ and $\beta :G\Rightarrow H$ are natural transformations $\mathcal{A}\to \mathcal{B}$, their vertical composite $\beta \circ \alpha :F\Rightarrow H$ has components $(\beta \circ \alpha {)}_A={\beta }_A\circ {\alpha }_A$.

Definition: Horizontal Composition (Whiskering)

If $\alpha :F\Rightarrow G:\mathcal{A}\to \mathcal{B}$ and $\beta :H\Rightarrow K:\mathcal{B}\to \mathcal{C}$, the horizontal composite $\beta \ast \alpha :HF\Rightarrow KG:\mathcal{A}\to \mathcal{C}$ has components $(\beta \ast \alpha {)}_A={\beta }_{GA}\circ H{\alpha }_A=K{\alpha }_A\circ {\beta }_{FA}$.

In particular, for a functor $H:\mathcal{B}\to \mathcal{C}$ and $\alpha :F\Rightarrow G:\mathcal{A}\to \mathcal{B}$, the whiskering $H\alpha :HF\Rightarrow HG$ has $(H\alpha {)}_A=H({\alpha }_A)$.

The Interchange Law

Vertical and horizontal composition satisfy the interchange law:

$$({\beta }^{\prime }\ast {\alpha }^{\prime })\circ (\beta \ast \alpha )=({\beta }^{\prime }\circ \beta )\ast ({\alpha }^{\prime }\circ \alpha )$$

whenever the composites make sense. This makes natural transformations the 2-cells of the 2-category CAT.

Connections

• Natural transformations are the morphisms of functor categories $[\mathcal{A},\mathcal{B}]$ (Functor Categories).
• The Yoneda lemma (Yoneda Lemma) characterises natural transformations $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$.
• Adjunctions (Adjoint Functors) can be defined as natural isomorphisms $\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$ or equivalently via natural transformations $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$.
• Limits are defined in terms of natural transformations (cones are natural transformations to a diagram) via General Limits.

See Also

• Functors — What natural transformations transform between
• Functor Categories — Category whose morphisms are natural transformations
• Yoneda Lemma — Fundamental theorem about natural transformations out of representables
• Adjoint Functors — Defined via unit/counit natural transformations