Timeline for Which direction of the adjoint functor theorem is most useful?
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| Feb 14, 2021 at 16:23 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 14, 2021 at 3:21 | comment | added | Mike Shulman | 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. | |
| Feb 14, 2021 at 1:28 | comment | added | Tim Campion | The situation is complicated by things like Lurie's proof of HTT 5.2.6.3 (proof that $P(f) : P(C) ^\to_\leftarrow P(C'): f^\ast$ is an adjunction for $f: C \to C'$), which still puzzles me -- he's already constructed the functors (with much effort in the case of $P(f)$), and yet he still chooses to verify the existence of an adjoint abstractly before proving that these particular functors are adjoint. Perhaps you're right and this is basically stylistic. | |
| Feb 14, 2021 at 1:24 | comment | added | Tim Campion | You may be right that the temptation to do so stems from the fact that folks are working with more complicated objects which are going to necessitate more abstract representability results anyway, so they end up not making too much of a distinction. | |
| Feb 14, 2021 at 1:23 | comment | added | Tim Campion | @MikeShulman I think I see -- basically all "representability" results end up getting thought of as "adjoint functor theorems" when talking about $\infty$-categorical techniques. Maybe it is worth making some distinctions. The need to construct higher coherence data makes proving the theorem "if every object has a universal arrow over $F$, then $F$ has an adjoint" (roughly HTT 5.2.4.2 I suppose) more difficult than in ordinary category theory, but maybe you're right that it's unhelpful to lump subsequent appeals to this theorem together with appeals to more abstract representability results. | |
| Feb 14, 2021 at 0:46 | comment | added | Mike Shulman | 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. | |
| Feb 14, 2021 at 0:43 | comment | added | Mike Shulman | 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. | |
| Feb 14, 2021 at 0:43 | comment | added | Mike Shulman | 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. | |
| Feb 13, 2021 at 20:39 | comment | added | Yemon Choi | 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 | |
| Feb 13, 2021 at 20:20 | history | answered | Tim Campion | CC BY-SA 4.0 |