2019/05/14 by Anno, Rina, Logvinenko, Timothy
#14F05 #18D20 #18E30 #18G99 #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.05740
We propose a new theory of (non-split) Pn-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of IdA 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 Pn-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 Pn-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 Pn-functors, we prove Segal's conjecture on their relation to spherical functors. Finally, we give four examples of non-split Pn-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.