Homotopy for functors

Question

I am reading this paper on Homotopy for functors by Ming-Jung Lee.

The author gives a definition (at the beginning of section $3$) as follows :

Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ be covariant functors of small categories. We say that $\varphi$ is homotopic to $\varphi'$ if there are covariant functors $\varphi_i:\Lambda\rightarrow \Gamma$ for $i=0,1,\cdots,n$ such that $\varphi_0=\varphi,\varphi_n=\varphi'$ and for each $i$, there is a natural transformation between $\varphi_i$ and $\varphi_{i+1}$.

Given two natural transformations, there is an obvious way to compose them which gives a natural transformation.

What is the point of considering the finite collection of natural transformations between given two functors. We can just define that two functors are homotopic if there is a natural transformation between them.

I do not understand what I am missing. Any suggestions are helpful.

Answer 1

The author means there is a zigzag of natural transformations. That is, "a natural transformation between $\varphi_i$ and $\varphi_{i+1}$" is intended to be nonspecific as to the direction of the transformation: it could go from $\varphi_i$ to $\varphi_{i+1}$ or from $\varphi_{i+1}$ to $\varphi_{i}$.

This is a reasonable notion of "homotopy" between functors because upon passing to geometric realizations / classifying spaces, any natural transformation induces a homotopy in the topological sense, and homotopies in the latter sense can always be reversed as well as composed; thus any zigzag of natural transformations between functors induces a single homotopy between their geometric realizations.

Answer 2

This may not be an answer but too long for a comment!

Though I did not read your mentioned paper https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf in details but what I understood from the Peter May's answer here Homotopy of functors that if you define a homotopy between 2 covariant functors $F,G :C \rightarrow D$ as a natural transformation between them then this notion of "homotopy" will not induce an expected equivalence relation on the set of functors from $C$ to $D$. So to solve this problem Ming-jung Lee defined a notion of "homotopy" between $F$ and $G$ as a sequence of covariant functors $\phi_1,....\phi_n:C \rightarrow D$ such that $\phi_1=F$ and $\phi_n=G$ and such that for each $i$ there exists a natural transformation between $\phi_i$ and $\phi_{i+1}$(where the direction of each natural transformation is unspecified). Also as mentioned in corollary 8 in https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf that if $F$ and $G$ are "homotopic" then the induced continuous maps $BF$ and $BG$ between geometric realisations $BC$ and $BD$ of the morphism complexes $MC$ and $MD$ respectively are also homotopic. So the notion of "homotopy" as mentioned by Ming-Jung Lee seems reasonable..