Homotopy of functors

Question

Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper Homotopy for Functors, definition 6:

If $\mathcal{C}$ and $\mathcal{D}$ are small categories and $\varphi, \varphi':\mathcal{C}\rightarrow\mathcal{D}$ are covariant functors, we say that $\varphi$ and $\varphi'$ are homotopic (written $\varphi\simeq\varphi'$) if there is a sequence of covariant functors $\varphi_1, ..., \varphi_n:\mathcal{C}\rightarrow\mathcal{D}$ such that $\varphi_1=\varphi$ and $\varphi_n=\varphi'$, and such that for each $i$ there is a natural transformation between $\varphi_i$ and $\varphi_{i+1}$. (Of course the direction of each natural transformation is unspecified.) In the linked paper, Lee associates to each small category $\mathcal{E}$ a simplicial set $M\mathcal{E}$ called its "morphism complex", and proves that the simplicial maps $M\varphi, M\varphi':M\mathcal{C}\rightarrow M\mathcal{D}$ induced by $\varphi$ and $\varphi'$ are homotopic as simplicial maps if $\varphi, \varphi'$ are homotopic as functors (and hence also the induced continuous maps between the geometric realizations of these simplicial sets are homotopic).

A second proposed definition comes from Ronnie Brown in section 6.5 of his book Topology and Groupoids:

Let $\mathbf{I}$ be the tree groupoid on two elements $0$ and $1$. Then if $\mathcal{C}$ and $\mathcal{D}$ are arbitrary categories and $\varphi, \varphi':\mathcal{C}\rightarrow\mathcal{D}$ are covariant functors, we say that $\varphi$ and $\varphi'$ are homotopic (also written $\varphi\simeq\varphi'$) if there is a functor $\Phi:\mathcal{C}\times\mathbf{I}\rightarrow\mathcal{D}$ such that the induced functors $\Phi(\_ , 0), \Phi(\_ , 1):\mathcal{C}\rightarrow\mathcal{D}$ equal $\varphi$ and $\varphi'$ respectively. (Here we abuse notation and denote $\Phi(\_ , id_0)$ and $\Phi(\_ , id_1)$ by $\Phi(\_ , 0)$ and $\Phi(\_ , 1)$.)

The book includes many results employing this definition; for instance, the fundamental groupoid functor $\pi:\mathbf{Top}\rightarrow\mathbf{Grpd}$ preserves homotopy: if $f, g:X\rightarrow Y$ are homotopic continuous maps, then the induced functors $\pi f, \pi g:\pi X\rightarrow\pi Y$ are homotopic as functors.

As another example, consider the "track groupoid" functor $\mathbf{Top}^{op}\times\mathbf{Top}\rightarrow\mathbf{Grpd}$, which takes a pair of spaces $(X, Y)$ to the groupoid $\pi Y^X$ that has as objects the continuous maps $f, g:X\rightarrow Y$ and morphisms the homotopy classes ($\mathrm{rel}$ end maps) of homotopies $F:f\simeq g$. One can show that this functor preserves homotopy in the same sense: if $f, f':Y\rightarrow W$ and $g, g':Z\rightarrow X$ are pairs of homotopic continuous maps, then the induced functors $\pi f^g, \pi f'^{g'}:\pi Y^X\rightarrow\pi W^Z$ are homotopic as functors. There are a number of other examples throughout the book.

It is easy to see that $\varphi$ and $\varphi'$ are homotopic under Brown's definition iff there is a natural equivalence between them, i.e. a natural transformation from $\varphi$ to $\varphi'$, each component of which is an isomorphism. In this formulation it is clear that Brown's definition is stronger than Lee's. My question is this: in which contexts is each definition more useful? What are some arguments to favor one over the other?

Answer 1

I'm surprised this has been up for days with nobody telling Atticus the obvious. Let $I$ be the unit interval category with two objects, $0$ and $1$, and one non-identity arrow $0 \to 1$. A natural transformation $F\to G$ between functors $\mathcal C \to \mathcal D$ is the same thing as a functor $\mathcal C \times I \to D$ that restricts to $F$ on $\mathcal C\times 0$ and to $G$ on $\mathcal C\times 1$. Of course, this notion of ``homotopy'' is not an equivalence relation, so the obvious thing to do is to take zigzags to make it into one, as Ming-Jung Lee does. Passage to classifying spaces then preserves homotopy since it preserves products and take $I$ to the topological unit interval $[0,1]$.

I apologize to the experts for saying the obvious, but it might help novices. This interpretation has been used, explicitly or implicitly, since very early on, I imagine well before 1972. Of course there is much more to say on a more advanced level, perhaps starting with Thomason's equivalence between the homotopy categories of (small) categories and of spaces.