A right semimodel structure on semisimplicial sets
Jan Rooduijn

Abstract:
In this thesis we investigate to what extent the Kan-Quillen model structure on simplicial sets can be transferred along the left adjoint of the free-forgetful adjunction with semisimplicial sets. We establish the novel result that, while the full model structure cannot be transferred, the underlying right semimodel structure can. Using earlier results we show that the adjunction becomes an equivalence between right semimodel structures, and that the fibrant objects and fibrations between fibrant objects are characterised by having the right lifting property against the semisimplicial horn inclusions. We show by a counterexample that this is not the case for all fibrations, for not every morphism that maps to a simplicial anodyne extension is a semisimplicial anodyne extension. Finally, we show that the strong anodyne extensions of simplicial and semisimplicial sets do coincide.