Natural transformations as categorical homotopies

Question

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible definition of natural transformation, which appears to be a categorification of homotopy:

given two functors $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ a natural transformation is a functor $\varphi \colon \mathcal C \times 2 \to \mathcal D$, where $2$ is the arrow category $0 \to 1$, such that $\varphi(-,0)=\mathcal F$ and $\varphi(-,1)=\mathcal G$.

My question is:

why doesn't anybody use this definition of natural transformation which seems to be more "natural" (at least for me)?

(Edit:) It seems that many people use this definition of natural transformation. This arises the following question:

Is there any introductory textbook (or lecture) on category theory that introduces natural transformation in this "homotopical" way rather then the classical one?

(Edit2:) Some days ago I've read a post in nlab about $k$-transfor. In particular I have been interested by the discussion in the said post, because it seems to prove that the homotopical definition of natural transformation should be the right one (or at least a slight modification of it). On the other end this definition have always seemed to be the most natural one, because historically category theory develop in the context of algebraic topology, so now I've a new question:

Does anyone know the logical process that took Mac Lane and Eilenberg to give their (classical) definition of natural transformation?

Here I'm interested in the topological/algebraic motivation that move those great mathematicians to such definition rather the other one.

Answer 1

The homotopy analogue definition of natural transformations has been known and used regularly since at least the late 1960's, by which time it was understood that the classifying space functor from (small) categories to spaces converts natural transformations to homotopies because it takes the category $I=2$ to the unit interval and preserves products. Composition of natural transformations $H\colon A\times I\to B$ and $J\colon B\times I\to C$ is just the obvious composite starting with $id\times \Delta: A\times I \to A\times I\times I$, just as in topology. (I've been teaching that for at least several decades, and I'm sure I'm not the only one.)

Answer 2

Once you learn a subject, you can think about things in whatever way is most pleasing or helpful for solving a problem. Fixing a fact as a definition is pedagogy -- something to help those learning the subject.

I can't really speak for how others learn, but I'm not sure recognizing natural transformations as being described by functors $\mathcal{C} \times 2 \to \mathcal{D}$ would be very useful before one starts seriously thinking in terms of the 2-category of categories.

I confess I would almost turn your question on its head -- I far more frequently want to think of a homotopy between functions $f,g:X \to Y$ as being a function from $X$ to paths in $Y$, or sometimes as a function from $[0,1]$ to $Y^X$, and feel the usual definition as a function $X \times [0,1] \to Y$ more as being a much simpler way to state the technical details. I saw the analogy with homotopy early in learning about categories, and I don't think seeing natural transformations defined as functors $\mathcal{C} \times 2 \to \mathcal{D}$ would have helped me make the analogy. (But, for the record, I am very much not an algebraic topologist)

Answer 3

Disclaimer: this is not an answer to the question as I have no explanation for why people don't introduce natural transformations in the way explained in the question, but I am posting this in order to expand a comment I made. The comment was

this is the starting observation to make for introducing simplicial categories as a model for $\infty$--categories

Moreover, I am not a specialist neither of category theory nor of homotopy theory (and a posteriori of higher categories).

The $2$-category of categories

The starting point is that the category $Cat$ of categories is actually a $2$-category. For any to objects (i.e. categories) $\mathcal C$ and $\mathcal D$ we have that $Hom_{Cat}(\mathcal C,\mathcal D)$ is itself a category.

This is very transparent when using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=\mathbb{G}^1$ is the arrow category $0\to 1$ and $t_{\leq0}(Cat)$ is the underlying $1$-category of $Cat$.

Remark: In general one can see a $2$-category $\mathcal C$ as a simplicial category by replacing the $Hom$-categories by their nerves.

In the case of $Cat$, we see that the $Hom$-categories naturally appear as $1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular").

The $3$-category of $2$-categories

Le us now go to natural transformations of (strict) $2$-functors between (strict) $2$-categories. Given two such $2$-functors $F,G:\mathcal C\to\mathcal D$ one can see that a natural transformation $F\Rightarrow G$ is the same as a $2$-functors $$ \phi:\mathcal C\times \mathbb{G}^2\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,1)=G$, where $\mathbb{G}^2$ is the $2$-category with two objects $0$ and $1$ and such that $Hom_{\mathbb{G}^2}(0,1)$ is the arrow category $\mathbb{G}^1=(0\to 1)$.

Therefore the "set" of $2$-functors is a naturally a $2$-category.

Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.

In the case of $2-Cat$, we see that the $Hom$-$2$-categories naturally appear as $2$-truncations of globular sets.

Simplices, Cubes, and globes

The globe category $\mathbb{G}$, the cubical category $\Box$ and the simplicial category $\Delta$ are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak) $\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak) $\infty$-groupoids.

I don't know any reference but I guess that the same holds for globular sets (which are quite more used by people working with automata).

The $(n+1)$-category of $n$-categories

Let me consider the category $n-Cat$ of (strict) $n$-categories. A a natural transformation between (strict) $n$-functor $F,G:\mathcal C\to\mathcal D$ can be written as an $n$-functor $$ \phi:\mathcal C\times \mathbb{G}^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,1)=G$, where $\mathbb{G}^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{\mathbb{G}^n}(0,1)$ is the $(n-1)$-category $\mathbb{G}^{n-1}$.

Therefore the "set" of $n$-functors is a naturally a (strict) $n$-category, and thus $n-Cat$ is a (strict) $n+1$-category. It also naturally appears as a $n$-truncation of a globular category.

The advantage of working with simplicial/cubical/globular categories

Working directly with simplicial/cubical/globular categories has the following advantages:

1. it does allow to work directly with higher categories without going through an inductive process.
2. it allows to deal with weak $(\infty,1)$-categories, as simplicial/cubical/globular are models for weak $\infty$-groupoids (here $(\infty,1)$ stands for "$\infty$-categories such that $n$-arrows for $n\geq2$ are weakly invertible").

Answer 4

This "geometric" definition is well-known to category-theorists. See for example this youtube video by the Catsters, which introduces natural transformations. It should be also well-known to algebraic topologists working with model categories. But I have to admit that there are few introductions to category theory which emphasize this definition of a natural transformation.

Remark that this fits into a more general framework: For every category $C$, there is an isomorphism $[I,C] \cong Arr(C)$, where $Arr(C)$ is the arrow category of $C$. In particular, $Arr([C,D]) \cong [I,[C,D]] \cong [C \times I,D]$.

On the other hand, the usual definition is more easy to work with. For example how do you define the composition of two natural transformations, say given by $\alpha : C \times 2 \to D, \beta : C \times 2 \to D$ with $\alpha(-,1) = \beta(-,0)$? Of course you can just write it down explicitly, but then you end up working with the usual definition. But instead, you could also use that $\alpha,\beta$ correspond to a functor on the amalgam $(C \times 2) \cup_C (C \times 2)$ of the inclusions $(-,1)$ and $(-,0)$, and compose with the natural functor $C \times 2 \to (C \times 2) \cup_C (C \times 2)$ which "leaves out the middle point".

Answer 5

Concerning

Have anyone ever introduced natural transformation in this "homotopical" way rather then the classical one in any reference like a textbook or some lecture notes?

Yes, Quillen introduces it in the paper

Higher Algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.

In connection with his "Theorem A" and "Theorem B."

Answer 6

What is "more natural" is strictly determined by a mathematical background one has (or more seriously --- by one’s understanding of the world) when one comes to learn a new subject. Thus, a good definition should be more about "simplicity" (with respect to its theory) than about "analogy" to other concepts (in other braches of math). Analogies are then established by theorems.

I am not a mathematician, so I have a sweet opportunity to be ignorant on some fundamental branches of math --- for example --- topology. I think of functors $\mathbb{C} \rightarrow \mathbb{D}$ as of structures in $\mathbb{D}$ of the shape of $\mathbb{C}$. Then a transformation is something that morphs one structure into another (i.e. it is a collection of morphisms indexed by the shape of a structure), whereas natural transformation is something that morphs in a coherent way.

I really like a story on "Blind men and an elephant" link text that I first red in Peter Johnstone's book "Sketches of an Elephant". He compares a topos to the elephant, and we are the blind men. Surely, we are blind men, but I do think that most concepts found in category theory (with perhaps category theory itself) are like elephants.

Answer 7

Charles Ehresmann had a natty way of developing natural transformations. For a category $C$ let $\square C$ be the double category of commuting squares in $C$. Then for a small category $B$ we can form Cat($B,\square_1 C$), the functors from $B$ to the direction 1 part of $\square C$. This gets a category structure from the category structure in direction 2 of $\square C$. So we get a category CAT($B,C$) of functors and natural transformations. This view makes it easier to verify the law

Cat($ A \times B,C) \cong $Cat($ A, $CAT($B,C$)).

And this method goes over to topological categories as well:

R. Brown and P. Nickolas, ``Exponential laws for topological categories, groupoids and groups and mapping spaces of colimits'', Cah. Top. G\'eom. Diff. 20 (1979) 179-198.

See also Section 6.5 of my book Topology and Groupoids for using the homotopy terminology for natural equivalences, as it was in the first 1968 edition entitled "Elements of Modern Topology" (McGraw Hill).

Answer 8

Following the previous indication of Professor Brown I want to add another possible way to see natural transformation which is a generalization of the previous definition.

Given categories $\mathcal C$ and $\mathcal D$ and two functors between them $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal D$ then a natural transformation $\tau$ can be defined as a functor $\tau \colon \mathcal C \to (\mathcal F \downarrow \mathcal G)$ which arrow components are the diagonal functions, sending each arrow $f \in \mathcal C(c,c')$, with $c,c' \in \mathcal C$ to $(f,f) \in (\mathcal F \downarrow \mathcal G)(\tau(c),\tau(c'))$.

Edit: I think the definition of natural transformation proposed by professor Brown probably can be even a more natural than the one proposed in the question. I think that more details are worthed.

The key ingredient for that definition is the concept of arrow category of a given category $\mathbf D$: such category have morphism of $\mathbf D$ as objects and commutative square as morphisms.

This category come equipped with two functors $\mathbf {source}, \mathbf{target} \colon \text{Arr}(\mathbf D) \to \mathbf D$ such that for each object (i.e. a morphisms of $\mathbf D$) $f \colon d \to d'$ we have $$\mathbf{source}(f)=d$$ $$\mathbf{target}(f)=d'$$ while for each $f \in \mathbf D(x,x')$, $g \in \mathbf D(y,y')$ and a morphism $\alpha \in \text{Arr}(\mathbf D)(f,g)$ (i.e. a quadruple $\langle f,g, \alpha_0,\alpha_1\rangle$ where $\alpha_0 \in \mathbf D(x,y)$ and $\alpha_1 \in \mathbf D(x',y')$ such that $\alpha_1 \circ f = g \circ \alpha_0$) we have $$\mathbf{source}(\alpha)=\alpha_0$$ $$\mathbf{target}(\alpha)=\alpha_1$$ it's easy to prove that these data give two functors (which gives to $\text{Arr}(\mathbf D)$ the structure of a graph internal to $\mathbf{Cat}$).

Now let's take a look to this new definition of natural transformation:

A natural transformation $\tau$ between two functors $F,G \colon \mathbf C \to \mathbf D$ is a functor $\tau \colon \mathbf C \to \text{Arr}(\mathbf D)$ such that $\mathbf{source} \circ \tau = F$ and $\mathbf{target}\circ \tau = G$.

A functor of this kind associate to every object $c \in \mathbf C$ a morphism $\tau_c \colon F(c) \to G(c)$ in $\mathbf D$, while to every $f \in \mathbf C(c,c')$ it gives the commutative triangle expressing the equality $$\tau_{c'} \circ F(f)=\tau_{c'} \circ \mathbf {source}(\tau_f)=\mathbf {target}(\tau_f) \circ \tau_c = G(f) \circ \tau_c$$ certifying the naturality (in the ordinary sense) of the $\tau_c$. This definition reminds the notion of homotopy between maps $f,g \colon X \to Y$ as map of kind $X \to Y^I$ (i.e. an homotopy as a (continuous) family of path of $Y$).

That's not all, indeed we can reiterate the construction of the arrow category obtaining what I think is called a cubical set $$\mathbf D \leftarrow \text{Arr}(\mathbf D) \leftarrow \text{Arr}^2(\mathbf D)\leftarrow \dots $$ where each arrow should be thought as the pair of functors $\mathbf{source}_{n+1},\mathbf{target}_{n+1} \colon \text{Arr}^{n+1}(\mathbf D) \to \text{Arr}^n (\mathbf D)$.

In this way we can associate to each category a cubical set. There's also a natural way to associate to every functor a (degree 0) mapping of cubical sets.

If we consider natural transformation as maps from a category to an arrow category then this correspondence associate to each natural transformation a degree 1 map between such cubical sets (by degree one I mean that the induced map send every object of $\text{Arr}^n(\mathbf C)$ in an object of $\text{Arr}^{n+1}(\mathbf D)$). I've found really beautiful this construction because it shows an analogy between categories-functors-natural transformation and complexes-map of complexes-complexes homotopies.