Simplicial sets are a co-reflective subcategory of cubical sets with connections via straightening, enabling transfer of cofibrantly generated model structures with monomorphisms as cofibrations.
Jardine, Categorical homotopy theory, Homology, Homotopy Appl
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A co-reflection of cubical sets into simplicial sets with applications to model structures
Simplicial sets are a co-reflective subcategory of cubical sets with connections via straightening, enabling transfer of cofibrantly generated model structures with monomorphisms as cofibrations.