I will introduce a candidate model for (infinity-)categories weakly enriched in chain complexes, based on the Joyal model for quasi-categories. The main idea is to replace the category of simplicial sets by a category of certain colax monoidal functors, inspired by a result of Leinster and Bacard's work on Segal dg-categories. We call them ``templicial k-modules'' and define ``quasi-categories in Mod(k)'' analogously to the classical situation. Equipping these with some extra structure, they become equivalent to non-negatively graded dg-categories through a lift of Lurie's dg-nerve.
At present, no analogue of Joyal's model structure exists for templicial k-modules, but I will outline some partial results in that direction. This is joint work with Wendy Lowen.
