topos theory

# Contents

## Definition

Let $A$ be a small category, and let $\mathrm{Psh}\left(A\right)={\mathrm{Set}}^{{A}^{\mathrm{op}}}$ be the category of presheaves on $A$. Since $\mathrm{Psh}\left(A\right)$ is a Grothendieck topos, it has a unique subobject classifier, $L$.

Let $0$ and $1$ denote the initial object and terminal object, respectively, of $\mathrm{Psh}\left(A\right)$. The presheaf $1$ has exactly two subobjects $0↪1$ and $1↪1$. These determine the unique two elements ${\lambda }^{0},{\lambda }^{1}\in L\left(1\right)=\mathrm{Hom}\left(1,L\right)$.

We call the triple $𝔏=\left(L,{\lambda }^{0},{\lambda }^{1}\right)$ the Lawvere interval for the topos $\mathrm{Psh}\left(A\right)$. 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 $\mathrm{Psh}\left(A\right)$, the object $L$ is fibrant.

###### Proof

Given any monomorphism $A\to B$ and any morphism $A\to L$, there exists a lifting $B\to L$.

To see this, notice that the morphism $A\to L$ classifies a subobject $C↪A$. However, composing this with the monomorphism $A↪B$, this monomorphism is classified by a morphism $B\to L$ making the diagram commute.

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

###### Proposition

Given any small set of monomorphisms in $\mathrm{Psh}\left(A\right)$, there exists the smallest Cisinski model structure for which those monomorphisms are trivial cofibrations.

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

## Examples

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