nLab Lawvere interval

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Contents

Definition

Let AA be a small category, and let Psh(A)=Set A opPsh(A)=Set^{A^{op}} be the category of presheaves on AA. Since Psh(A)Psh(A) is a Grothendieck topos, it has a unique subobject classifier, LL.

Let 0\mathbf{0} and 1\mathbf{1} denote the initial object and terminal object, respectively, of Psh(A)Psh(A). The terminal presheaf 1\mathbf{1} has two distinguished subobjects 01\mathbf{0}\hookrightarrow \mathbf{1} and 11\mathbf{1}\hookrightarrow \mathbf{1}, which correspond to global points λ 0,λ 1L(1)=Hom(1,L)\lambda^0,\lambda^1\in L(\mathbf{1})=Hom(\mathbf{1},L) of the subobject classifier.

Definition

The triple 𝔏=(L,λ 0,λ 1)\mathfrak{L}=(L,\lambda^0,\lambda^1) is called the Lawvere interval for the topos Psh(A)Psh(A). This object determines a cylinder functor given by taking the cartesian product with LL, called the Lawvere cylinder.

This is due to Cisinski (2006), §1.3.9.

By Prop. below, the Lawvere interval 𝔏\mathfrak{L} may be regarded as the universal cylinder object for Cisinski model structures on presheaf toposes.

Properties

The Lawvere cylinder is universal in the sense that any LL-homotopy lifts to a homotopy for any other good cylinder object.

Proposition

Let AA be an object of a topos. Given a diagram

A⨿A(δ 0,δ 1)CσA A \amalg A \stackrel{(\delta^0, \delta^1)}{\to} C \stackrel{\sigma}{\to} A

where the copies of AA are disjoint subobjects of CC splitting σ\sigma, then there is a morphism of cylinders CL×AC \to L \times A.

Proof

Let χ i:CL\chi^i : C \to L be the characteristic function for δ i\delta^i. That the copies of AA are disjoint subobjects implies χ 0¬χ 1\chi^0 \leq \neg \chi^1, and so χ 0χ 1χ 1¬χ 1\chi^0 \vee \chi^1 \leq\chi^1 \vee \neg \chi^1.

Since xx¬xx \mapsto x \vee \neg x is the characteristic function of λ:1⨿1L\lambda : 1 \amalg 1 \to L, the morphism (χ 1,σ):CL×A(\chi^1, \sigma) : C \to L \times A pulls A×λA \times \lambda back to a subobject containing δ\delta. Compatibility with σ\sigma is obvious, and so it is a morphism of cylinder objects.

Monomorphisms and trivial fibrations

In a topos, lifting problems against monomorphisms can be described in terms of partial maps, giving an elementary characterization of trivial fibrations:

Proposition

Let p:XSp : X \to S. Suppose the partial map classifier XOpt S(X)X \hookrightarrow \mathrm{Opt}_S(X) exists in the slice category over SS. Then Opt S(X)\mathrm{Opt}_S(X) represents the presheaf (in AA) of diagrams of the form modulo the identification of equivalent subobjects BAB \subseteq A.

Corollary

If p:XSp : X \to S is an arrow in a topos E\mathbf{E}, it has the right lifting property against monomorphisms iff the restriction X×LOpt S(X)X \times L \to \mathrm{Opt}_S(X) is a split epi.

Proof

X×LX \times L represents the presheaf (in AA) of all such diagrams that have a filler AXA \to X, since XAX \to A determines the lower triangle and from that, XLX \to L determines the upper triangle.

Thus pp has the claimed property iff XLOpt S(X)X \to L \to \mathrm{Opt}_S(X) represents an epimorphism of presheaves, which is equivalent to being a split epi.

Proposition

If p:XSp : X \to S is an arrow in a topos E\mathbf{E}, it has the right lifting property against monomorphisms iff the insertion XOpt S(X)X \hookrightarrow \mathrm{Opt}_S(X) has a retraction.

Proof

By the properties of partial map classifiers, every lifting problem has a unique factorization where the left square is a pullback. So pp has the right lifting property against the mono XOpt S(X)X \to \mathrm{Opt}_S(X) iff it has the right lifting property against all monos.

Corollary

If p:XSp : X \to S is an arrow in a topos E\mathbf{E}, then XOpt S(X)S X \hookrightarrow \mathrm{Opt}_S(X) \to S is a factorization into a monomorphism followed by a morphism with the right lifting property against monomorphisms.

Proof

In any topos (in particular E /S\mathbf{E}_{/S}), the both inclusions Opt(X)Opt(Opt(X))\mathrm{Opt}(X) \hookrightarrow \mathrm{Opt}(\mathrm{Opt}(X)) are split by the operation of taking the intersection of domains.

The argument spelled out in the case of X=1X = 1 and Opt(X)=L\mathrm{Opt}(X) = L is spelled out in

Proposition

The subobject classifier in any topos is an injective object, whence the Lawvere interval LL (Def. ) is a fibrant resolution of the terminal object in any Cisinski model structure on Psh(A)Psh(A).

This is highlighted at the end of Cisinski (2006), §1.3.9 with reference (up to a typo) to MacLane Moerdijk (1992), IV §10.1:
Proof

Recall that for LL 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 L*L \to \ast is an acyclic cofibration.

Hence assuming the solid diagram above, we show the existence of f̲\underline{f}:

Here f:BLf \colon B \to L classifies a subobject of BB, which we denote CBC \hookrightarrow B. The point now is that, with BAB \hookrightarrow A being a monomorphism, the composite

CBA C \hookrightarrow B \hookrightarrow A

exhibits CC also as a subobject of AA, which as such is classified by some map f̲:AL\underline{f} \colon A \to L, and we claim that this serves as the desired extension. To see that indeed this f̲\underline{f} makes the above triangle commute, consider the following commuting diagram:

Here

  • the right rectangle is the pullback square that witnesses f̲\underline{f} as the classifying map of CAC \hookrightarrow A, by definition,

  • the left rectangle is the fiber product B ACB \cap_A C computed via the pasting law for pullbacks to be isomorphic to CC, using that CC is already a subobject of AA (which gives the factorization in the middle) and then using (for the two squares on the left) that:

    1. the fiber product of a monomorphism with itself is its domain (this Prop.),

    2. 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 CC. But CC was defined to be classified by ff, and so the uniqueness of subobject classifying maps implies that the total bottom map equals ff, which was to be shown.

Anodyne maps and fibrant objects

For the Lawvere interval in a presheaf topos, Cisinski’s anodyne completion operation stops at the first step Λ L(,mono)=Λ L 0(,mono)\Lambda_L(\varnothing, \mathrm{mono}) = \Lambda_L^0(\varnothing, \mathrm{mono}) which is given by pushout products of monomorphisms with the endpoints 1L\mathbf{1} \to L, due to the associativity and symmetry of the pushout product. This leads to

Definition

In a topos, say that an object is (𝔏\mathfrak{L}-)-fibrant iff both endpoint evaluations X LXX^L \to X are trivial fibrations.

Proposition

In a presheaf topos E\mathbf{E}, the following are equivalent for an arrow p:XSp : X \to S:

  1. pp is a fibrant object of E /S\mathbf{E}_{/S}

  2. X LS L× SXX^L \to S^L \times_S X is a trivial fibration, for both endpoints 1L\mathbf{1} \to L.

  3. pp is a naive 𝔏\mathfrak{L}-fibration in the Cisinski model structure

  4. pp has the right lifting property against 𝔏\mathfrak{L}-anodyne maps in the Cisinski model structure

Proof

(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 pp by 1L\mathbf{1} \to L has the right lifting property against a monomorphism ii iff pp has the right lifting property against the pushout product of ii with 1L\mathbf{1} \to L, by Joyal-Tierney calculus. So (2) is equivalent to (3) and (4).

(1) is the assertion that p1 Sp \to 1_S is a naive 𝔏 S\mathfrak{L}_S-fibration. The generating anodyne morphisms there are the pushout products (AB)× S¯(SS×L)(A \subseteq B) \overline{\times_S} (S \to S \times L). But this is just the pushout product (AB)ׯ(1L)(A \subseteq B) \overline{\times} (1 \to L) equippped with a map BSB \to S. So this is equivalent to pp being a naive 𝔏\mathfrak{L}-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 X LOpt X(X L)X^L \to \mathrm{Opt}_X(X^L) 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.

References

Last revised on July 20, 2024 at 05:04:36. See the history of this page for a list of all contributions to it.