The Unapologetic Mathematician

Natural Transformations and Functor Categories

So we know about categories and functors describing transformations between categories. Now we come to transformations between functors — natural transformations.

This is really what category theory was originally invented for, and the terminology predates the theory. Certain homomorphisms were called “natural”, but there really wasn’t a good notion of what “natural” meant. In the process of trying to flesh that out it became clear that we were really talking about transformations between the values of two different constructions from the same source, and those constructions became functors. Then in order to rigorously define what a functor was, categories were introduced.

Given two functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{C}\rightarrow\mathcal{D}$, a natural transformation $\eta:F\rightarrow G$ is a collection of arrows $\eta_C:F(C)\rightarrow G(C)$ in $\mathcal{D}$ indexed by the objects of $\mathcal{C}$. The condition of naturality is that the following square commutes for every arrow $f:A\rightarrow B$ in $\mathcal{C}$:

$\begin{matrix}F(A)&\rightarrow^{\eta_A}&G(A)\\\downarrow^{F(f)}&&\downarrow^{G(f)}\\F(B)&\rightarrow^{\eta_B}&G(B)\end{matrix}$

The vertical arrows from from applying the two functors to the arrow $f$, and the horizontal arrows are the components of the natural transformation.

If we have three functors $F$, $G$, and $H$ from $\mathcal{C}$ to $\mathcal{D}$ and natural transformations $\eta:F\rightarrow G$ and $\xi:G\rightarrow H$ we get a natural transformation $\xi\circ\eta:F\rightarrow H$ with components $(\xi\circ\eta)_C=\xi_C\circ\eta_C$. Indeed, we can just stack the naturality squares beside each other:

$\begin{matrix}F(A)&\rightarrow^{\eta_A}&G(A)&\rightarrow^{\xi_A}&H(A)\\\downarrow^{F(f)}&&\downarrow^{G(f)}&&\downarrow^{H(f)}\\F(B)&\rightarrow^{\eta_B}&G(B)&\rightarrow^{\xi_B}&H(B)\end{matrix}$

and the outer square commutes because both the inner ones do.

Every functor comes with the identity natural transformation $1_F$, whose components are all identity morphisms. Clearly it acts as the identity for the above composition of natural transformations.

A natural transformation is invertible for the above composition if and only if each component is invertible as an arrow in $\mathcal{D}$. In this case we call it a “natural isomorphism”. We say two functors are “naturally isomorphic” if there is a natural isomorphism between them.

All of this certainly looks like we’re talking about a category, but again the set theoretic constraints often work against us. There are, however, times where we really do have a category. If one of $\mathcal{C}$ or $\mathcal{D}$ (or both) are small, then all the set theory works out and we get an honest category of functors from $\mathcal{C}$ to $\mathcal{D}$. We will usually denote this category as $\mathcal{D}^\mathcal{C}$. Its objects are functors from $\mathcal{C}$ to $\mathcal{D}$, and its morphisms are natural transformations between such functors.

And now we can explain the notation $\mathcal{C}^\mathbf{2}$ for the category of arrows. This is the category of functors from $\mathbf{2}$ to $\mathcal{C}$! What is a functor from $\mathbf{2}$ to $\mathcal{C}$? Remember that $\mathbf{2}$ is the category with two objects $A$ and $B$ and one non-trivial arrow $f:A\rightarrow B$. Thus a functor $F:\mathbf{2}\rightarrow\mathcal{C}$ is defined by an arrow $F(f):F(A)\rightarrow F(B)$, and there’s exactly one functor for every arrow in $\mathcal{C}$.

Now let’s say we have two such functors $F(f):F(A)\rightarrow F(B)$ and $G(f):G(A)\rightarrow G(B)$. A natural transformation $\eta:F\rightarrow G$ consists of morphisms $\eta_A:F(A)\rightarrow G(A)$ and $\eta_B:F(B)\rightarrow G(B)$ so that the naturality square commutes. But this is the same thing we used to define morphisms in the arrow category, just with some different notation!

Natural transformations and functor categories show up absolutely everywhere once you know to look for them. We’ll be seeing a lot more examples as we go on.

Like this:

Posted by John Armstrong | Category theory

7 Comments	»

1. [...] we start with some small category that describes the form of a diagram, and then we take the category of functors into the category we’re interested in [...]

Pingback by | June 16, 2007 | Reply

2. [...] natural transformation between two functors from to picks out a morphism in for each object in , subject to a [...]

Pingback by | August 17, 2007 | Reply

3. [...] us to the category of such functors. The objects, recall, are functors, while the morphisms are natural transformations. Now let’s consider what, exactly, a natural transformation consists of in this [...]

Pingback by | October 28, 2008 | Reply

4. [...] are also deep connections between -morphisms and natural transformations, in the categorical viewpoint. Those who are really interested in that can dig into the archives a [...]

Pingback by | September 21, 2010 | Reply

5. [...] should note that these are not just isomorphisms, but “natural” isomorphisms. That the construction is a functor is clear, and it’s straightforward to verify that these [...]

Pingback by | October 11, 2010 | Reply

6. [...] yes. Any two vector spaces having the same dimension are isomorphic, but they’re not “naturally” isomorphic. Roughly, there’s no universal method of giving an explicit isomorphism, and so it’s [...]

Pingback by | October 13, 2010 | Reply

7. [...] opposite way, and then said a presheaf of sets is a functor . So the natural home for them is the functor category , where the morphisms are natural [...]

Pingback by | March 19, 2011 | Reply

« Previous | Next »

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 46, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.939337968826294, "perplexity_flag": "head"}