For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1,\dots ,A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1}\to A_i\left(D\right)$, where D is the distinguished invertible object of $\mathcal{C}$.

We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L}={\int }^X{X}^{\vee }\otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995).

Our main motivation stems from the application to finite dimensional quasi-Hopf algebras H. For the category of finite-dimensional H-modules, we relate the four monadic cointegrals (two of which require H to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called γ-symmetrised cointegrals in the pivotal case, for γ the modulus of H.

For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of S and T which uses the monadic cointegral.

对于 $\mathcal{C}$ 一个有限张量类别，我们考虑中央单子的四个版本， ${\mathrm{一种}}_{\mathrm{1个}}，\dots ，{\mathrm{一种}}_4$ 上 $\mathcal{C}$。其中有两个是霍普夫（Hopf）单子，$\mathcal{C}$至关重要，其余两个也是关键。在那种情况下${\mathrm{一种}}_{\mathrm{一世}}$与Hopf monads同构。我们为 ${\mathrm{一种}}_{\mathrm{一世}}$ 成为 ${\mathrm{一种}}_{\mathrm{一世}}$模态 $\mathbf{1个}\to {\mathrm{一种}}_{\mathrm{一世}}（d）$，其中D是的独特的可逆对象$\mathcal{C}$。

我们将单子共积分与清水（2019）引入的分类共积分相关联，以防万一 $\mathcal{C}$ 编织到编织Hopf代数的积分 $\mathcal{大号}={\int }^X{X}^{\vee }\otimes X$ 在 $\mathcal{C}$ Lyubashenko（1995）研究。

我们的主要动机源于对有限维拟霍夫代数H的应用。对于有限维H模的类别，我们将四个单子共积分（其中两个需要H为枢轴）与准霍夫代数的四个现存协积分概念相关：Hausser和Nill（ 1994年），以及在关键情况下所谓的γ对称共积分，对于γ，H的模量。

对于（不一定是半简单的）模块化张量类别 $\mathcal{C}$，柳巴申科（Lyubashenko）在 $\mathcal{C}$，尤其是 $\mathrm{小号}\mathrm{大号}（2，\mathbb{ž}）$ 上 $\mathcal{C}（\mathcal{大号}，\mathbf{1个}）$。在可分解带状拟霍夫代数的情况下，我们给出使用单子共积分的S和T的作用的简单表达式。