Nerves of enriched categories via necklaces

Abstract We present a unified treatment for simplicial nerves of enriched categories by means of necklaces, which we call “necklicial nerves”. This includes the homotopy coherent, differential graded and cubical nerves. It is shown that every necklicial nerve can be lifted to templicial objects, an enriched variant of simplicial sets introduced by Wendy Lowen and the present author. Further, we give sufficient conditions for a necklicial nerve to be a quasi‐category. Inspired by the work of Dugger and Spivak on the rigidification of quasi‐categories ( Algebr. Geom. Topol . 11(2011), 263–325), 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.