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?

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    $\begingroup$ I think that both of these references are more or less obsolete. The first notion you're referring to is fleshed out in detail in the Asterisque 301 volume by Georges Maltsiniotis. Note that this all lives in a world where categories model spaces. The other construction does not make sense in general and uses the homotopy structure on Top in an essential way. In that case, take a look at the appendices of Higher Topos Theory or Hirschhorn's book in the section on simplicial model categories. $\endgroup$ – Harry Gindi Dec 30 '19 at 4:40
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    $\begingroup$ It is a case of horses for courses. In the case of functors to groupoids or strict higher groupoids, then the approach of my book, and its sequel, Nonabelian Algebraic Topoology, , is often convenient. $\endgroup$ – Ronnie Brown Dec 30 '19 at 15:23
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    $\begingroup$ @Harry Grindi I should say that T&G still seems to be the only algebraic topology book in English to use the idea of a set of base points for use in descent situations re Van Kampen's theorem, as strongly supported by Grothendieck in his Esquisse. Whether the simplicial approach is the "best" is another matter. Some motivitic homotopy work uses cubical sets with connections, for technical reasons that the simplicial did not work. See my 2018 paper in Indag. $\endgroup$ – Ronnie Brown Dec 30 '19 at 17:01

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.

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  • $\begingroup$ Peter thank you so much for the answer; is this the canonical choice then? (And in particular also why favor this formulation over Brown's condition of natural equivalence between the two functors? Despite being stricter that condition also seems flexible enough to preserve homotopy in quite a few desirable situations, as listed above.) $\endgroup$ – Atticus Stonestrom Jan 2 at 0:20
  • $\begingroup$ Also apologies if the question is too elementary for mathoverflow; what references would you recommend for understanding Thomason's result? $\endgroup$ – Atticus Stonestrom Jan 2 at 0:21
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    $\begingroup$ The analogy with topology is too close for this to be anything but canonical. I have nothing against Ronnie's choice, but it is not in common use. The most recent paper about Thomason's theorem and its generalizations is available on my web page as math.uchicago.edu/~may/PAPERS/MSZ.pdf, which gives background and references. No apologies needed. A good opportunity to advertise a classical observation that still seems much less well-known than it should be. $\endgroup$ – Peter May Jan 2 at 5:07

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