Cofree coalgebras over operads

This paper explicitely constructs cofree coalgebras over operads in the category of DG-modules. Special cases are considered in which the general expression simplifies (such as the pointed, irreducible case). It is shown that the existence of an operad-action on a coalgebra implies a ``generalized coassociativity'' that facilitates the construction. Some areas of application are discussed.Comment: A revised version of this paper will appear in the journal Topology and its Applications (communicated by Paul Goerss). The revision involves editorial changes to the manuscrip

Quiver Bialgebras and Monoidal Categories

We study the bialgebra structures on quiver coalgebras and the monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed monoidal categories.Comment: 10 page

Localization in tame and wild coalgebras

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations".Comment: 23 pages, to appear in Journal of Pure and Applied Algebr