Local Complete Segal Spaces

We show that the complete Segal model structure extends to a model structure on bimplicial presheaves on a small site $\mathscr{C}$, for which the weak equivalences are local (or stalkwise) weak equivalences. This model structure can be realized as a left Bousfield localization of the Jardine model structure on the simplicial presheaves on a site $\mathscr{C}/ \Delta^{op}$. Furthermore, it is shown that this model structure is Quillen equivalent to the model structure of the author's previous preprint entitled 'the Local Joyal Model Structure'. This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney.