A cofibrantly generated model structure on a topos, for the cofibrations are exactly the monomorphisms, is called a Cisinski model structure. Cofibrantly generated means that there are small sets and of morphisms, on which the small object argument can be applied, so that they generate all cofibrations and trivial cofibrations using the lifting property:[2]
More generally, a small set generating the class of monomorphisms of a category of presheaves is called cellular model:[3][4]
Joyal model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions and acyclic cofibrations (inner anodyne extensions) are generated by inner horn inclusions (with and ).[6][7]
Kan–Quillen model structure: Cofibrations (monomorphisms) are generated by the boundary inclusions and acyclic cofibrations (anodyne extensions) are generated by horn inclusions (with and ).[6]