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Let k be a commutative ring. We study the behavior of coverings of k-categories through fiber products and find a criterion for a covering to be Galois or universal. [ABSTRACT FROM AUTHOR]

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures between a finite set algebra and the grouplike coalgebra on two elements. The resulting 2-nilpotent algebras have particular features with respect to Hochschild (co)homology and cyclic homology. [ABSTRACT FROM AUTHOR]

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt(DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived. [ABSTRACT FROM AUTHOR]