nLab
Lawvere interval

Context

Topos Theory

Could not include topos theory - contents

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 presheaf 11 has exactly two subobjects 01\mathbf{0}\hookrightarrow \mathbf{1} and 11\mathbf{1}\hookrightarrow \mathbf{1}. These determine the unique two elements λ 0,λ 1L(1)=Hom(1,L)\lambda^0,\lambda^1\in L(\mathbf{1})=Hom(\mathbf{1},L).

We call the triple 𝔏=(L,λ 0,λ 1)\mathfrak{L}=(L,\lambda^0,\lambda^1) the Lawvere interval for the topos Psh(A)Psh(A). This object determines a unique cylinder functor given by taking the cartesian product with an object. We will call this endofunctor the Lawvere cylinder .

Properties

Proposition

With respect to the Cisinski model structure on Psh(A)Psh(A), the object LL is fibrant.

Proof

Given any monomorphism ABA\to B and any morphism ALA\to L, there exists a lifting BLB\to L.

To see this, notice that the morphism ALA\to L classifies a subobject CAC\hookrightarrow A. However, composing this with the monomorphism ABA\hookrightarrow B, this monomorphism is classified by a morphism BLB\to L making the diagram commute.

For this reason, 𝔏\mathfrak{L} can be considered the universal cylinder object for Cisinski model structures on a presheaf topos.

Proposition

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

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

Examples

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.

Revised on December 8, 2010 14:48:25 by Urs Schreiber (131.211.233.8)