Local Complete Segal Spaces

TL;DR

This paper extends the complete Segal model structure to bimplicial presheaves on a site, establishing a local model structure and a Quillen equivalence to the Local Joyal Model Structure, generalizing known equivalences.

Contribution

It introduces a local complete Segal model structure on bimplicial presheaves and proves its Quillen equivalence to the Local Joyal Model Structure, extending existing Segal-Joyal equivalences.

Findings

Established a local model structure on bimplicial presheaves.

Proved the model structure is a left Bousfield localization.

Demonstrated Quillen equivalence with the Local Joyal Model Structure.

Abstract

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.