Nerves of enriched categories via necklaces

Abstract

We introduce necklicial nerve functors from enriched categories to simplicial sets, which include Cordier's homotopy coherent, Lurie's differential graded and Le Grignou's cubical nerves. It is shown that every necklicial nerve can be lifted to the templicial objects of arXiv:2302.02484v2. Building on the work of Dugger and Spivak, we give sufficient conditions under which the left-adjoint of a necklicial nerve can be described more explicitly. As an application, we obtain novel and simple expressions for the left-adjoints of the dg-nerve and cubical nerve.