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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.

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