The 1-Segal and 2-Segal Model Structures

Abstract

In this chapter, we introduce the notions of 1-Segal and 2-Segal objects in a combinatorial model category C. If further C admits the structure of a left proper, tractable, symmetric monoidal model category, then we introduce model structures for 1-Segal and 2-Segal objects which arise as enriched Bousfield localizations of the injective model structure on C _Δ. For ${\mathbf C} = {\mathbb S}$, the model structure for 1-Segal objects in ${\mathbb S}$ recovers the Rezk model structure for 1-Segal spaces introduced in Rezk (Trans Am Math Soc 353(3):973–1007, 2001).