nLab Lawvere interval



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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.


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.



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:

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:


  • 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.


Given any small set of monomorphisms in Psh(A)Psh(A), there exists the smallest Cisinski model structure for which those monomorphisms are acyclic cofibrations.

By applying a theorem of Denis-Charles Cisinski. (…)


Suppose A=ΔA=\Delta is the simplex category, and let SS consist only of the inclusion {1}:Δ 0Δ 1\{1\}:\Delta^0\to\Delta^1. Applying Cisinski’s anodyne completion of SS by Lawvere’s cylinder Λ 𝔏(S,M)\mathbf{\Lambda}_\mathfrak{L}(S,M), 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.