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In the daily life of a working mathematician which direction of the adjoint functor theorem is more useful? Unpacking, does one find it more useful to:

a) prove that a functor admits an adjoint and conclude that it preserves limits/colimits,

OR

b) prove that a functor preserves limits/colimits and conclude that it admits an adjoint?

I guess I should also include a third option:

c) neither a) nor b) is true in general, it really depends on what part of math you work in.

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    $\begingroup$ FWIW, in my experience usually when people say "the adjoint functor theorem" they are referring only to (b). Property (a) isn't generally given a special name. $\endgroup$
    – Mike Shulman
    Jan 24, 2021 at 22:56
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    $\begingroup$ Steve Awodey taught me (a) as the mnemonic "RAPL", probably because it sounds better than "LAPC". $\endgroup$
    – Andrej Bauer
    Jan 24, 2021 at 23:24
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    $\begingroup$ Regarding the actual question, my guess would be that (a) is used more often than (b). But I'm not sure how anyone could give a definitive answer. $\endgroup$
    – Mike Shulman
    Jan 25, 2021 at 2:00
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    $\begingroup$ Usually when someone asks on MO whether a functor has an adjoint and the answer is not obvious, I reflexively go for the contrapositive of (a). $\endgroup$
    – Todd Trimble ♦
    Jan 25, 2021 at 2:36
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    $\begingroup$ If you can show that a functor preserves limits in some simple cases, that is empirical evidence suggesting that you try to find the right adjoint. The existence of the adjoint in general is far from trivial and may depend on the Axiom-Scheme of Replacement. I can't imagine that anyone would rely on that route as the public proof of existence. Besides, a construction is always better, even if you're classical. $\endgroup$
    – Paul Taylor
    Jan 27, 2021 at 17:50

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In 1-category theory, the easy direction (a) is invoked all the time. The hard direction (b) doesn't have to be formally invoked very often, because most adjoints can be constructed by hand (and even if you do initially construct an adjoint via the Adjoint Functor Theorem, usually you'll want to get a more explicit understanding of it as you move forward anyway). However, (b) is still used all the time as a heuristic consideration: if you want to know whether a functor admits an adjoint and it's not immediately obvious how to construct one, usually the thing to do is to check limit/colimit preservation.

In $\infty$-category theory, the situation is different (EDIT: At least superficially? Perhaps more deeply? See Mike Shulman's important objections in the comments below). There, (a) is still just as important, but (b) (in various incarnations) is invoked quite frequently. The reason is that in $\infty$-category theory, it is often difficult to construct functors explicitly! This is because it doesn't suffice to say what the functor does on objects and morphisms and check a funcotriality condition -- rather, you've got to specify higher coherence data all the way up. Adjoint functor-type theorems are used as ready-made packages where all of this coherence data can be supplied automatically. This is a central insight of Lurie and really one of the major factors making the whole theory useful.

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    $\begingroup$ Nice point! On a slight tangent, i.e. not related to AF theorems per se: even at 1-categorical level, for people like me it is sometimes useful to know that one can automatically get a left adjoint by constructing initial objects in appropriate comma categories (which then become the unit for the given adjunction). E.g. it seemed to go unremarked for a while in Banach-algebra world that the forgetful functor from "dual Banach algebras" to Banach algebras has a left adjoint, but this follows immediately from the universal property of a certain construction by Runde + the machinery I mentioned $\endgroup$
    – Yemon Choi
    Feb 13, 2021 at 20:39
  • $\begingroup$ I'm not convinced that this is a difference intrinsic to the 1/$\infty$ distinction. In 1-category theory I think almost any adjoint functor that can be constructed "entirely by hand" can equally be constructed by an operation sending each object to a universal arrow, and that doesn't seem like it should be significantly more difficult in $\infty$-category theory: you don't need to specify all the higher coherences of the adjoint functor, only check that certain maps on hom-spaces are equivalences. $\endgroup$
    – Mike Shulman
    Feb 14, 2021 at 0:43
  • $\begingroup$ So if people in $\infty$-category theory are really using the adjoint functor theorem more, it seems to me more likely to be a stylistic choice rather than a necessity. $\endgroup$
    – Mike Shulman
    Feb 14, 2021 at 0:43
  • $\begingroup$ Or due to some difference between 1 and $\infty$ other than the need for higher coherences, e.g. the relative inexplicitness of the objects of $\infty$-categories making it harder to even give an explicit construction of a universal arrow. $\endgroup$
    – Mike Shulman
    Feb 14, 2021 at 0:46
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    $\begingroup$ The Adjoint Functor Theorem proper is a theorem about the existence of objects with certain universal properties, which are constructed by combining the opposite kind of universal property (limit vs colimit) together with a "smallness" or "solution-set" condition. A theorem that starts with "if every object has a universal arrow" is not an AFT, because it's assuming the existence of such objects. $\endgroup$
    – Mike Shulman
    Feb 14, 2021 at 3:21

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