Weak Projections onto a Braided Hopf Algebra

We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.12. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra $A$ is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that $A$ is a double cross product, or biproduct in Madjid's terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial

Cotensor Coalgebras in Monoidal Categories

We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an abelian monoidal category. Under some further conditions we show that such a cotensor coalgebra exists and satisfies a meaningful universal property. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth

Small Bialgebras with a Projection: Applications

In this paper we continue the investigation started in [A.M.St.-Small], dealing with bialgebras $A$ with an $H$-bilinear coalgebra projection over an arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as deformed bosonizations R#_{\xi} H of a pre-bialgebra $R$ by $H$ with a cocycle $\xi$. Here we describe the behavior of $\xi$ in the case when $R$ is f.d. and thin i.e. it is connected with one dimensional space of primitive elements. This is used to analyze the arithmetic properties of $A$. Meaningful results are obtained when $H$ is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations (e.g. the multiplication of $R$ is not $H$-colinear or $\xi$ is non-trivial)

Braided Bialgebras of Type One

Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly $\mathbb{N}$-graded both as an algebra and as a coalgebra

The Heyneman-Radford Theorem for Monoidal Categories

We prove Heyneman-Radford Theorem in the framework of Monoidal Categories

Categories of comodules and chain complexes of modules

Let \lL(A) denote the coendomorphism left $R$-bialgebroid associated to a left finitely generated and projective extension of rings $R \to A$ with identities. We show that the category of left comodules over an epimorphic image of \lL(A) is equivalent to the category of chain complexes of left $R$-modules. This equivalence is monoidal whenever $R$ is commutative and $A$ is an $R$-algebra. This is a generalization, using entirely new tools, of results by B. Pareigis and D. Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the non commutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.Comment: The title has been changed, the first part is removed and the construction of the coendomorphim bialgebroid is now freely used in the statement of the main Theorem