Lawvere interval

*Could not include topos theory - contents*

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

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

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

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

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 $C\hookrightarrow A$. However, composing this with the monomorphism $A\hookrightarrow B$, this monomorphism is classified by a morphism $B\to L$ making the diagram commute.

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

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

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

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