A general formalism produces directed model structures from a cylinder object, unifying the covariant and contravariant model structures on simplicial sets and the Cartesian and coCartesian model structures on marked simplicial sets.
Higher categories and homotopical algebra , volume 180 of Cambridge studies in advanced mathematics
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Covariant & Contravariant Homotopy Theories
A general formalism produces directed model structures from a cylinder object, unifying the covariant and contravariant model structures on simplicial sets and the Cartesian and coCartesian model structures on marked simplicial sets.