nLab
Lawvere interval

Definition

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

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

We call the triple $𝔏=(L,{\lambda }^0,{\lambda }^1)$ the Lawvere interval for the topos $\mathrm{Psh}(A)$. 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 $\mathrm{Psh}(A)$, 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 $C\hookrightarrow A$. 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, $𝔏$ can be considered the universal cylinder object for Cisinski model structures on a presheaf topos.

Proposition

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

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

Examples

Suppose $A=\Delta$ 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 ${\Lambda }_{𝔏}(S,M)$, we get exactly the contravariant model structure on the category of simplicial sets.