We introduce a notion of enriched infinity categories over a given monoidal category, in analogy with quasi-categories over the category of sets. We make use of certain colax monoidal functors, which we call templicial objects, as a variant of simplicial objects that respects the monoidal structure. We relate the resulting enriched quasi-categories to nonassociative Frobenius monoidal functors, allowing us to prove that the free templicial module over an ordinary quasi-category is a linear quasi-category. To any dg category we associate a linear quasi-category, the linear dg nerve, which enhances the classical dg nerve, and we argue that linear quasi-categories can be seen as relaxations of dg-categories.

Joint work with Arne Mertens.