| Let C a category and T a monad. This paper mainly talks about some propositions and results of algebras and coalgebras in categories, which can be extended to those of monads and comonads. It makes some interesting results. It is composed of five chapters.Chapter one is the introduction which introduces the background of monad and comonad and the ways used.Chapter two is preparatory knowledge, which mainly introduces the definitions of algebra, coalgebra, monad, comonad, functors, natural transformation, adjoint functor, cotensor product and so on, it also gives an equivalent proposition about adjoint functors in [1].Chapter three gives the definitions of cosemisimple comonad and cosep-arable comonad, then it gives two propositions equivalent to the definition separately.Chapter four gives the definitions of the entwined structure and entwined modules between monads and comonads. It also gives two examples which extends the entwined structure and entwined modules of algebras and coalgebras. Furthermore it builds two functors between categories of entwined modules, then it proves them are adjoint functors.In chapter five firstly it defines two categories of bicomodule, between them it builds two functors, and proves they have adjoint functors separately. Then it defines comonads over firm monads, and gets equivalent propositions to coseparable comonads. Lastly it defines coderivations and cointegrations of T-comonads and builds an isomorphism between abelian groups of them. |
