Frobenius templicial modules and the dg-nerve

Abstract

Templicial objects were put forth in arXiv:2302.02484v2 to set up a suitable simplicial framework for enriched quasi-categories. Following Leinster, these objects feature certain comultiplications as a replacement for outer face maps in the non-cartesian case. In the present paper, we consider Frobenius templicial objects, thus re-introducing multiplications into the picture. When enriching over $k$-modules for a commutative ring $k$, we prove an equivalence of categories between (homologically) positively graded dg-categories on the one hand and Frobenius templicial modules on the other hand. This equivalence yields a natural enrichment of the classical dg-nerve, turning dg-categories into quasi-categories in modules. Assuming a projectivity condition, we further prove that a templicial module is a quasi-category in modules precisely when it can be equipped with a nonassociative Frobenius structure.