The construction of cofree coalgebras

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Everyone knows that the cofree coalgebra exists ‘on general principles’. As far as constructions go, most attention has been given to vector spaces, and these depend on the idea of duals of free algebras on duals (although the pointed irreducible case is straightforward). We here offer two simple constructions of the cofree algebra generated by a module over a commutative ring, with applications to the cohomology of bialgebras.