Hochschild Cohomology, Monoid Objects and Monoidal Categories

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories.

Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bilgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomolo...

Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras

Let (M,⊗,1) be an abelian monoidal category. In order to investigate the structure of Hopf algebras with Chevalley property (i.e. Hopf algebras having the coradical a Hopf subalgebra) we define the Hochschild cohomology of an algebra A in M. Then we characterize those algebras A which have dimension less than or equal to 1 with respect to Hochschild cohomology. We use these results to prove tha...

Monoidal Categories, 2-traces, and Cyclic Cohomology

In this paper we show that to a unital associative algebra object (resp. co-unital coassociative co-algebra object) of any abelian monoidal category (C,⊗) endowed with a symmetric 2-trace, i.e. an F ∈ Fun(C,Vec) satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in F...

Cohomology of Monoids in Monoidal Categories