Hochschild cohomology and “smoothness” in monoidal 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

Hochschild and Ordinary Cohomology Rings of Small Categories

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C| = BC. We study the ring homomorphism HH∗(kC) → H∗(|C|, k) and prove it is split surjective, using the factorization category of Quillen [16] and certain techniques from functor cohomology theory. This generalizes the well-known results for finite groups...