UC Irvine

In this paper, we begin with the bar construction of a (noncommutative) dg-algebra. We go over the concept of a Hirsch associative algebra, turning the bar construction into a bialgebra. We move on to the bar construction of a module over that algebra. Using the Hirsch algebra, we introduce a twisted tensor product in order to construct a tensor product for left modules over our algebra and show that in the case when our algebra is commutative, our tensor product is quasi-isomorphic to the Tor functor. We introduce the concepts of a dg-nerve of a category and monoidal ∞-categories and use these constructions as guidelines to prove that left modules over a Hirsch associative form a monoidal ∞-category.