Triple adjoints and anti-involutions

Philip Hackney – 29 November 2017

Often, one can transfer a (cofibrantly-generated) model structure by lifting fibrations and weak equivalences along a right adjoint. We give a concise sufficient condition for lifted model structures to exist when the right adjoint has a right adjoint of its own.

This condition applies to give model structures on * simplicially-enriched categories equipped with an anti-involution, and * simplicial sets equipped with an anti-involution lifted from the Bergner and Joyal model structures, respectively. Moreover, the usual Quillen equivalence between these lifts to this setting.