TITLE: Is every functor inducing a homotopy equivalence a composition of adjoint functors?
QUESTION [8 upvotes]: It was asked here whether every functor is a composition of adjoint functors. The answer is no, because all adjoint functors induce homotopy equivalences on the nerve, and we can construct functors that do not induce homotopy equivalences.
My question is the following: can all functors inducing homotopy equivalences on the nerve be expressed as compositions of adjoint functors?

REPLY [12 votes]: The answer is no.  Let $C$ be a category such that the unique map from $C$ to the terminal category is a composition of $n$ adjoints.  Then $C$ has an object $x_0$ such that every other object of $C$ can be connected to $x_0$ by a zigzag of length at most $n$; this is easy to prove by induction on $n$.  
In particular, let $R$ be the "infinite zigzag", the unique poset such that $|BR|$ is homeomorphic to $\mathbb{R}$.  Then $BR$ is contractible, but the unique map from $R$ to the terminal category cannot be a composition of adjoints.  Note, however, that this map is a transfinite composition of adjoints (of length $\omega$).  It seems plausible to me that any functor between (small) categories which is an equivalence on nerves could be a transfinite composition of adjoints.