Bar Simplicial Modules and Secondary Cyclic (Co)homology

In this paper we study the simplicial structure of the complex $C^{\bullet }((A,B,\epsilon );M)$, associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object $\mathcal{B}(A,B,\epsilon )$, which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 3.9 and 4.11).