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Alonso Perez-Lona (Sep 29 2026 at 14:19):In Theorem 6.10 of "Generalized stability for abstract homotopy theories" by Groth and Shulman 1704.08084, it is shown that as soon as the adjoint 7-tuple of a pointed derivator extends further, even just by one additional functor no matter if to the left or to the right, one automatically gets an infinite number of adjoints on both sides and effectively enters the stable context. I find this very confusing which indicates to me I am just not understanding the meaning behind the proof. This is particularly confusing given how one can think of the category of spectrum objects of a pointed -category with finite limits as a homotopy limit of -categories . This makes me think I can choose to truncate this sequence wherever I want and obtain a big but finite number of adjoints in a corresponding derivator.
What is a good intuition that helps one see why this expectation is wrong?