{\em Galois comodules} over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of {\em Galois functors} over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context.
Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a {\em grouplike natural transformation} $g:I\to G$ generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to comonoidal functors on monoidal categories in the sense of A. Bruguières and A. Virelizier. As well-know, for any set $G$ the product $G\times-$ defines an endofunctor on the category of sets and this is a Hopf monad if and only if $G$ allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to {\em Galois objects} in the sense of Chase and Sweedler.
Gerstenhaber type structures on Davydov-Yetter cohomology with coefficients
Indian Institute of Technology, Gandhinagar（印度理工学院甘地纳加尔分校）；Indian Institute of Science, Bangalore（印度科学学院班加罗尔分校）；Indian Institute of Technology, Delhi（印度理工学院德里分校）
2-范畴中的弱混合分配律（也称为弱缠绕结构）由单子和余单子以及以推广Beck的混合分配律的方式联系它们的2-胞腔组成。我们证明了弱混合分配律可以描述为单子和余单子的相容对，在2-范畴中，分别扩展了余单子的2-范畴和单子的2-范畴。在此基础上，我们定义了一个2-范畴，它的0-胞腔是弱混合分布律。在一个2-范畴K中，它允许单子和余单子的Eilenberg-Moore构造，并且其中幂等元2-胞分裂，我们从这个2-范畴的弱混合分配律到K^{2 x 2}构造一个完全忠实的2-函子。
