Reference for factorization of left adjoints?

Question

Some time ago, in connection with trying to understand a construction of Amitsur (Embeddings of matrix rings, Pac JM 36 (1971), I stumbled across a short paper which considered some variant of the following question:

given (forgetful) functors $H: {\mathcal C} \to {\mathcal B}$ and $G:{\mathcal B}\to {\mathcal A}$, such that $GH:{\mathcal C}\to {\mathcal A}$ and $G:{\mathcal B}\to {\mathcal A}$ both have left adjoints, when can we deduce that $H$ has a left adjoint?

I think in the intended application ${\mathcal A}$ was the category of sets and ${\mathcal B}$ was the category of rings; the idea being that it is easier or more transparent to construct "the free widget on a generating set" than "the free widget on a generating ring".

Unfortunately, I can't remember the name of the author nor the title of the paper. Does this sound familiar to anyone? and if not, does anyone have an alternative reference? Sorry to ask such a vague question, but I have been racking my brains and searching on MathSciNet to no avail.

UPDATE: while reading Todd Trimble's answer below, the veil inexplicably lifted and I was able to remember the author's surname, from which a quick Google turned up what I was after:

An Interpolation Theorem for Adjoint Functors S. A. Huq, Proceedings of the American Mathematical Society Vol. 25, No. 4 (Aug., 1970), pp. 880-883

Sorry to waste people's time!

Answer 1

This should probably be a comment but I don't know how to include links in comments.

Results of this type are often called adjoint triangle theorems. There are many such: see John Power's paper A unified approach to the lifting of adjoints for a summary (and a unified approach).

A sufficient condition for $H$ to admit a left adjoint is that $C$ has coequalizers and that $G$ is ``of descent type''. (This means that if we form the monad $S$ on $A$ induced by $G$ and its left adjoint, then the induced functor $B\to A^S$ is fully faithful.) This includes the important examples of the type considered in Todd's answer.

for a summary.

Answer 2

"Relatively free" functors have been considered at least since the time of Lawvere's thesis; see for example page 111 of 122 for the case involving finitary algebraic theories.

I don't know where this is written down, but the following construction is pretty general and might suit your purposes. Let $\theta: S \to T$ be a morphism of monads on a category $C$, and suppose that the category of algebras $C^T$ has coequalizers (as happens if for example $C= Set$). Then the forgetful functor

$$C^T \to C^S,$$

which takes a $T$-algebra $(d, \alpha: Td \to d)$ to the $S$-algebra

$$Sd \stackrel{\theta d}{\to} Td \stackrel{\alpha}{\to} d,$$

has a left adjoint. This gives the factorization in the evident case where $\mathcal{A} = C$, $\mathcal{B} = C^S$, and $\mathcal{C} = C^T$.

The left adjoint takes an $S$-algebra $(c, \xi: Sc \to c)$ to the coequalizer of the pair of arrows in $C^T$:

$$T\xi: TSc \to Tc, \qquad TSc \stackrel{T\theta c}{\to} TTC \stackrel{mc}{\to} Tc,$$

where $m: TT \to T$ is the multiplication on $T$. This coequalizer might be written $T \circ_S c$, by analogy with the coequalizer $B \otimes_A M$ where $B$ is an algebra over $A$ and $M$ is an $A$-module.

That this is the left adjoint is a slightly lengthy diagram chase which I can produce if desired. (I think I may have written this up in the nLab somewhere, but just now I can't access the nLab. It might be at a page on algebras over a monad, or on free algebras. If it's not there, I should record the detailed proof at the Lab when I get a chance.)

Edit: I wrote up something quickly here at the nLab, which has a detailed (somewhat pedestrian) proof of the adjunction stated above.