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Adjoint functor theorem involves solution set condition and somehow sneaks in some category of elements or comma categories. I hate this way of doing things.

Is there any alternate way of doing it, say via Kan extension?

Adjoints can be expressed as Kan extensions,

$\operatorname{Ran}_{\mathcal{G}} I_{\mathcal{B}} \cong \mathcal{F}\dashv \mathcal{G}\cong \operatorname{Lan}_{\mathcal{F}} I_{\mathcal{A}}.$

I was thinking existence of adjoints can be translated into existence of some special Kan extensions, then for the coend formula to make sense, certain conditions will have to be met, this would turn out to be the solution set condition somehow?

Viśvāmitra
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  • "I hate this way of doing things." Why? Let me make a guess: you believe that somehow this involves talking about elements and therefore goes against the spirit of category theory. This is false: it involves talking about collections of objects which have co/limits. Arbitrary collections do not have co/limits, but it is common that all small enough collections (aka sets) have co/limits. This does not involve talking about elements of objects in your category. – Pilcrow Oct 07 '22 at 20:48
  • @Pilcrow Yes, it doesn't involve talking about elements of objects, but it's still in the same spirit. Maybe 'higher set theory'. In this case, we want to say if there is a reverse direction arrow, with certain properties, talking using objects in a category seems like talking about elements of sets to construct some function between sets. – Viśvāmitra Oct 08 '22 at 05:52
  • @MjrPajeet It's not more and no less problematic than talking about products in a category, which involves taking a product of a set of objects. – Pilcrow Oct 12 '22 at 09:37
  • @Pilcrow, yes perhaps, it's also probably unavoidable, but I was just looking for some way to hide it, or make it look more natural in some sense. But I like fosco's answer below. That's the sort of motivation I was looking for, it kind of motivates as to what we are trying to do. – Viśvāmitra Oct 17 '22 at 10:05

1 Answers1

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The problem -in short, and quite roughly speaking- is that you need some assumptions ensuring that the colimit/coend to compute the left Kan extension $Lan_F1$ exists, as the domain might be a large category.

Conditions like the solution set condition or other forms of adjoint functor theorem are usually of the form "the colimit you would compute in order for the adjoint to exist can actually be computed on a small category, so it does exist".

fosco
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  • These would be pointwise Kan extensions no? Are there any adjoint functor theorems where the conditions are not about ensuring colimits/coend existence? I mean to say, the situations where $Lan_F 1$ is not a coend but something not so well behaved? I don't know anything beyond the basics, so was wondering. – Viśvāmitra Oct 07 '22 at 19:35
  • Every time the adjunction exists you can characterize the right adjoint as the left extension of the identity along the left adjoint, and this extension is even more than pointwise, it is absolute. The converse implication ("if I have $Lan_F1$, then $F\dashv Lan_F1$") requires the extension to exists in the first place! – fosco Oct 08 '22 at 08:37
  • (The most genuine answer to your question then is: no, if $F\dashv Lan_F1$ then $Lan_F1$ must be an absolute extension) – fosco Oct 08 '22 at 08:38