Wedge Products and Cotensor Coalgebras in Monoidal Categories

DOI10.1007/S10468-008-9089-2arXivmath/0602016MaRDI QIDQ6476654

Alessandro Ardizzoni

Publication date: 1 February 2006

Abstract: The construction of the cotensor coalgebra for an "abelian monoidal" category $M$ which is also cocomplete, complete and AB5, was performed in [A. Ardizzoni, C. Menini and D. c{S}tefan, emph{Cotensor Coalgebras in Monoidal Categories}, Comm. Algebra, to appear]. It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra $E$ in $M$ is filled by considering a direct limit $widetildeD$ of a filtration consisting of wedge products of a subcoalgebra $D$ of $E$. The main aim of this paper is to characterize hereditary coalgebras $widetildeD$, where $D$ is a coseparable coalgebra in $M$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $widetildeD$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T_D^c(DwD/D)$ if and only if it is a cotensor coalgebra $T_D^c(N)$, where $N$ is a certain $D$-bicomodule in $M$. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.

Mathematics Subject Classification ID

Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) (18A30) Hopf algebras and their applications (16T05)

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