We call the triple the Lawvere interval for the topos . This object determines a unique cylinder functor given by taking the cartesian product with an object. We will call this endofunctor the Lawvere cylinder .
With respect to the Cisinski model structure on , the object is fibrant.
Given any monomorphism and any morphism , there exists a lifting .
To see this, notice that the morphism classifies a subobject . However, composing this with the monomorphism , this monomorphism is classified by a morphism making the diagram commute.
For this reason, can be considered the universal cylinder object for Cisinski model structures on a presheaf topos.
By applying a theorem of Denis-Charles Cisinski. (…)
Suppose is the simplex category, and let consist only of the inclusion . Applying Cisinski’s anodyne completion of by Lawvere’s cylinder , we get exactly the contravariant model structure on the category of simplicial sets.