Rewriting and presentations of quasicategories

Abstract

We show that the methods of rewriting theory to establish coherence theorems can applied at the level of quasicategories. More precisely, for any category C which admits a presentation by a convergent rewrite system, we show that the corresponding weak (infinity,1)-category admits an Amick-Groves-Squier style presentation whose generators correspond to (some of) the critical branchings of the rewrite system. This follows entirely from reinterpreting K. Brown's simplicial proof of the usual homological Anick-Groves-Squier presentation in terms of the Joyal model structure instead of the Kan-Quillen model structure. We give several applications of this to the theory of quasicategories, including a relatively general coherence theorem for loop-free planar category theoretic diagrams and a new proof that pushout of Dwyer maps are homotopy pushouts.