Local Complete Segal Spaces

We develop a model structure on bimplicial presheaves on a small site C, for which the weak equivalences are local ( or stalkwise) weak equivalences in the complete Segal model structure. We call this the local Complete Segal model structure. This model structure can be realized as a left Bousfield localization of the Jardine ( injective) model structure on the simplicial presheaves on a site C /.op. Furthermore, it is shown that this model structure is Quillen equivalent to themodel structure of the author's paper ( Meadows in TAC31( 24): 690711, 2016). This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney ( Categories in algebra, geometry and mathematical physics, contemporary mathematics, vol. 431. American Mathematical Society, Providence, pp 277- 326, 2007). As an application, we compare the notion of descent in the local Joyal model structure to the notion of descent in the injective model structure. Interestingly, this is a consequence of the Quillen equivalence between the local Joyal and local Complete Segal model structures.