Cohomology and homotopy
In higher category theory
Let be a small category, and let be the category of presheaves on . Since is a Grothendieck topos, it has a unique subobject classifier, .
Let and denote the initial object and terminal object, respectively, of . The presheaf has exactly two subobjects and . These determine the unique two elements .
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 .
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.
Given any small set of monomorphisms in , there exists the smallest Cisinski model structure for which those monomorphisms are trivial cofibrations.
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.