Monadic cointegrals and applications to quasi-Hopf algebras

For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C. Two them are Hopf monads, and for pivotal, so remaining two. In that case all Ai isomorphic as monads. We define monadic cointegral to be an Ai-module morphism 1→Ai(D), where D is distinguished invertible object relate cointegrals categorical introduced by Shimizu (2019), and, in braided, integral braided algebra L=∫XX∨⊗X studied Lyubashenko (1995). Our main motivation stems from application dimensional quasi-Hopf algebras H. finite-dimensional H-modules, (two which require H pivotal) existing notions algebras: usual left/right Hausser Nill (1994), well so-called γ-symmetrised pivotal case, γ modulus (not necessarily semisimple) modular categories C, gave actions surface mapping class groups certain Hom-spaces particular SL(2,Z) C(L,1). factorisable ribbon algebra, give simple expression action S T uses cointegral.