The construction of cofree coalgebras (Q1208226)

The definition of the cofree coalgebra is completely dual to that of the free algebra. The author offers two simple constructions of the cofree coalgebra generated by a module over a commutative ring. The first construction uses the recursive function approach while the second one looks more like the dual of a tensor algebra. Different types of free coalgebras (coassociative, cocommutative or Lie coalgebra) are obtained as subcoalgebras of the most general cofree nonassociative coalgebra. The cohomology of coalgebras is defined using a simplicial complex generated by repeated applications of the functor \(S\) of the cofree coalgebra. The bialgebra cohomology groups are defined via a double complex which uses both the free algebra and the cofree coalgebra functors \(T\) and \(S\). This construction works in the more general context of formal bialgebras over a triple \(T\) and a cotriple \(S\) connected by a distributive law \(\lambda\). [See also \textit{T. F. Fox} and \textit{M. Markl}, Distributive laws and the cohomology. In preparation (1994).] The usual coalgebra can be considered as a bialgebra over the category of sets equipped with the data \((T, S, \lambda)\) of such a type. This gives a certain cohomology theory for coalgebras.