# Proving the Special Adjoint Functor Theorem from the general Adjoint Functor Theorem

Often, when dealing with adjoint functor theorems, people go about proving each one separately, from first principles if you will (this is the course taken in MacLane). However, the names suggest there is a deeper connection between the two.

These are the relevant theorems:

• AFT: For $D$ a complete locally small category, a functor $G:D\to C$ has a left adjoint iff it preserves small limits and satisfies the solution set condition (for each object $c$ in $C$ we have a weakly universal set from $c$ to $G$)

• SAFT (general form): Let $D$ be a complete locally small category with a cogenerating set and let there exist pullbacks of classes of subobjects of objects in $D$. Let $C$ be locally small. A functor $G:D \to C$ has a left adjoint iff it preserves small limits and pullbacks of classes of monics.

• SAFT (classical form): Let $D$ be a complete, well-powered, locally small category with a cogenerating set. Let $C$ be locally small. A functor $G:D \to C$ has a left adjoint iff it preserves small limits.

I wonder if it is possible to deduce SAFT from AFT; can we, given the hypotheses of SAFT, in some way apply AFT?

Of course this is trivially true. One simply goes through the standard proof of SAFT, at some point forgets what he is doing, produces some universal arrows and applies AFT. This isn't really interesting.

What is interesting is that we can produce a nontrivial proof of SAFT in the case that the categories involved are well-powered (so we can nontrivially prove the classical form of SAFT). You can, broadly speaking, get a solution set at an object $c$ as the set of arrows of the form $c \to G r$, where $r$ ranges over the subobjects of a certain product of the elements of the cogenerating set. But the proof doesn't translate to a general setting.

My question is this: can AFT be applied in a nontrivial way to prove the general form of SAFT? Based on the hypotheses of SAFT, can we construct nontrivial solution sets?

I'm thinking we can't. To construct a solution set, we need to already have some set, from which we can then take elements, and local smallness doesn't seem to be enough here. Also, MacLane doesn't mention this variation on the proof, but does mention the classical case.

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Does anyone remember names of theorems like this? It's like "first" and "second" isom. of group theory, for which I assume nobody actually remembers what name goes with what basic fact. So...why not state the actual theorems you have in mind? (I tend to doubt that at this level of general nonsense anything nontrivial could actually happen.) –  BCnrd May 24 '10 at 21:18
Concerning AFT and SAFT, the terminology is standard. You can look it up in Mac Lane. –  Martin Brandenburg May 24 '10 at 21:32
Martin, with the help of Google I've sort of figured out what AFT and SAFT are saying. But these special names still puzzle me for the following reason: AFT is Freyd's criteria for existence of adjoint, discovered by him in 1963 or so, yet the criteria looks exactly the same as what Grothendieck knew and used in late 1950's to prove pro-representability results (for deformation functors, etale $\pi_1$, etc.): form a suitable "limit" with enough rigidity to ensure that it works. So what's an idea in Freyd's work that Grothendieck missed? –  BCnrd May 25 '10 at 4:29
I'm not familiar with Grothendieck's results, so I can't answer your question. I have expanded the question, to hopefully make it clearer. And for the record, I very much enjoy knowing names to theorems, even if those names aren't generally well known (which isn't the case here). –  Miha Habič May 27 '10 at 6:31
@Miha: I agree that it's nice to have names for theorems. What I meant with my initial comment was to ask for a statement (or indication of) the content of the theorems in addition to their names. Moot point. If you get around to checking out Grothendieck's method of construction of fundamental groups in sga1 (or other reference), you'll see the whole technique of Freyd's proof sitting right there. I had no idea that this method is attributed to someone other than Grothendieck! I learned what I know about category theory "on the street", so I'm ignorant of the culture of the field. –  BCnrd May 28 '10 at 2:30