nLab
Lawvere interval

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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)