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My understanding of Ben's answer to this question is that even though associated graded is not an adjoint functor, it's not too bad because it is a composition of a right adjoint and a left adjoint.

But are such functors really "not that bad"? In particular, is it true that any functor be written as the composition of a right adjoint and a left adjoint?

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up vote 41 down vote accepted

The answer is no, because the nerve functor turns an adjoint pair of functors between categories into inverse homotopy equivalences between spaces (this is because of the existence of the unit and counit and the fact that nerve turns natural transformations into homotopies). In particular, this means that any functor whose nerve is not a homotopy equivalence cannot be a composite of adjoints. For a very simple example, you could take the functor from the 2-object discrete category to the terminal category.

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A similar but simpler argument: any adjunction between categories $C$ and $D$ induces a bijection between their sets (or classes) $\pi_0 C$ and $\pi_0 D$ of connected-components. So categories with different numbers of connected-components are never linked by a chain of adjoint functors. Eric's example will also do here. – Tom Leinster Aug 16 '12 at 15:02

Here's a really trivial way to see that the answer is "no": a functor from the empty category to a nonempty category is never a composite of adjoints (since a functor from the empty category to a nonempty category is never an adjoint).

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