The Joyal model structure on simplicial sets is extended to a model structure on the simplicial presheaves on a small site, in which the cofibrations are monomorphisms and the weak equivalences are local (or stalkwise) Joyal equivalences. The model structure is shown to be left proper.
Keywords: simplicial presheaves, quasi-categories, model structure duality, trace, derivator, absolute colimit
2010 MSC: 18G30, 18F20, 55U35
Theory and Applications of Categories, Vol. 31, 2016, No. 24, pp 690-711.