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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 Lawvere cylinder is universal in the sense that any -homotopy lifts to a homotopy for any other good cylinder object.
Let be an object of a topos. Given a diagram
where the copies of are disjoint subobjects of splitting , then there is a morphism of cylinders .
Let be the characteristic function for . That the copies of are disjoint subobjects implies , and so .
Since is the characteristic function of , the morphism pulls back to a subobject containing . Compatibility with is obvious, and so it is a morphism of cylinder objects.
In a topos, lifting problems against monomorphisms can be described in terms of partial maps, giving an elementary characterization of trivial fibrations:
Let . Suppose the partial map classifier exists in the slice category over . Then represents the presheaf (in ) of diagrams of the form modulo the identification of equivalent subobjects .
If is an arrow in a topos , it has the right lifting property against monomorphisms iff the restriction is a split epi.
represents the presheaf (in ) of all such diagrams that have a filler , since determines the lower triangle and from that, determines the upper triangle.
Thus has the claimed property iff represents an epimorphism of presheaves, which is equivalent to being a split epi.
If is an arrow in a topos , it has the right lifting property against monomorphisms iff the insertion has a retraction.
By the properties of partial map classifiers, every lifting problem has a unique factorization where the left square is a pullback. So has the right lifting property against the mono iff it has the right lifting property against all monos.
If is an arrow in a topos , then is a factorization into a monomorphism followed by a morphism with the right lifting property against monomorphisms.
In any topos (in particular ), the both inclusions are split by the operation of taking the intersection of domains.
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.
For the Lawvere interval in a presheaf topos, Cisinski’s anodyne completion operation stops at the first step which is given by pushout products of monomorphisms with the endpoints , due to the associativity and symmetry of the pushout product. This leads to
In a topos, say that an object is (-)-fibrant iff both endpoint evaluations are trivial fibrations.
In a presheaf topos , the following are equivalent for an arrow :
is a fibrant object of
is a trivial fibration, for both endpoints .
is a naive -fibration in the Cisinski model structure
has the right lifting property against -anodyne maps in the Cisinski model structure
(3) and (4) restate the fact the anodyne maps and naive fibrations are constructed as a weak factorization system in Cisinski’s construction.
The pullback power of by has the right lifting property against a monomorphism iff has the right lifting property against the pushout product of with , by Joyal-Tierney calculus. So (2) is equivalent to (3) and (4).
(1) is the assertion that is a naive -fibration. The generating anodyne morphisms there are the pushout products . But this is just the pushout product equippped with a map . So this is equivalent to being a naive -fibration.
In a locally presentable topos, the small object argument is used to construct the (anodyne map, naive fibration) factorization system, and in particular to construct fibrant replacements.
The condition of being a a fibrant object is equivalent to asking to have a retraction, or equvialently for a solution to the lifting problem
In a presheaf topos, the left vertical map will turn out to be a trivial cofibration. This gives a concrete construction for the small argument to be applied to in order to construct fibrant replacements.
Last revised on July 20, 2024 at 05:04:36. See the history of this page for a list of all contributions to it.