${\mathbb{P}}^n$-functors

Abstract

We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segal's conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist' formula.

80 pages; v3; the introduction updated to make clear the main new results of this paper and to explain the necessity of working with DG enhancements

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Cited by in corpus (4)

• Adjunction in 2-categories
• Spherical functors and the flop-flop autoequivalence
• On some examples of spherical functors related to flops
• Hochschild cohomology and deformations of $\mathbb{P}$-functors