Cofree coalgebras over operads. (English) Zbl 1032.18004

Let \(\mathcal V\) be an operad in the category of DG-modules, \(C\) an \(R\)-free DG-module, where \(R\) is a commutative ring, and \(G\) a coalgebra over \(\mathcal V\). Then \(G\) is called the cofree coalgebra cogenerated by \(C\) if: (1) there is a morphism \(\varepsilon: G\to C\) of coalgebras over \(\mathcal V\), called the cogeneration map; (2) for any morphism \(f:D\to C\) of coalgebras over \(\mathcal V\) there exists a unique morphism \(\widehat f:D\to G\) of coalgebras over \(\mathcal V\) such that \(f=\varepsilon\circ\widehat f\).
In this paper the author gives an explicit construction of the cofree coalgebra over \(\mathcal V\). This construction is dual, but more difficult, to the construction of the free algebra over an operad. Special cases of coalgebras, such as pointed or irreducible ones are also considered.


18D50 Operads (MSC2010)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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