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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 terminal presheaf has two distinguished subobjects and , which correspond to global points of the subobject classifier.
The triple is called the Lawvere interval for the topos . This object determines a cylinder functor given by taking the cartesian product with , called the Lawvere cylinder.
By Prop. below, the Lawvere interval may be regarded as the universal cylinder object for Cisinski model structures on presheaf toposes.
The subobject classifier in any topos is an injective object, whence the Lawvere interval (Def. ) is a fibrant resolution of the terminal object in any Cisinski model structure on .
Recall that for to be an injective object means that every solid span as below, where the vertical map is a monomorphism, admits a dashed lifting as shown:
Since in a Cisinski model structure, by definition, the monomorphisms are precisely the cofibrations, injective objects here are equivalently those for which the terminal map is an acyclic cofibration.
Hence assuming the solid diagram above, we show the existence of :
Here classifies a subobject of , which we denote . The point now is that, with being a monomorphism, the composite
exhibits also as a subobject of , which as such is classified by some map , and we claim that this serves as the desired extension. To see that indeed this makes the above triangle commute, consider the following commuting diagram:
Here
the right rectangle is the pullback square that witnesses as the classifying map of , by definition,
the left rectangle is the fiber product computed via the pasting law for pullbacks to be isomorphic to , using that is already a subobject of (which gives the factorization in the middle) and then using (for the two squares on the left) that:
the fiber product of a monomorphism with itself is its domain (this Prop.),
isomorphisms are preserved by pullback (this Prop.).
This implies, again by the pasting law, that the total square is a pullback, hence that the total bottom map classifies . But was defined to be classified by , and so the uniqueness of subobject classifying maps implies that the total bottom map equals , which was to be shown.
Given any small set of monomorphisms in , there exists the smallest Cisinski model structure for which those monomorphisms are acyclic 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.
Last revised on June 11, 2023 at 11:20:15. See the history of this page for a list of all contributions to it.