# nLab Lawvere interval

Contents

topos theory

## Theorems

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of $(\infty,1)$-categories

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

for stable $(\infty,1)$-categories

for $(\infty,1)$-operads

for $(n,r)$-categories

for $(\infty,1)$-sheaves / $\infty$-stacks

# Contents

## Definition

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 terminal presheaf $\mathbf{1}$ has two distinguished subobjects $\mathbf{0}\hookrightarrow \mathbf{1}$ and $\mathbf{1}\hookrightarrow \mathbf{1}$, which correspond to global points $\lambda^0,\lambda^1\in L(\mathbf{1})=Hom(\mathbf{1},L)$ of the subobject classifier.

###### Definition

The triple $\mathfrak{L}=(L,\lambda^0,\lambda^1)$ is called the Lawvere interval for the topos $Psh(A)$. This object determines a cylinder functor given by taking the cartesian product with $L$, called the Lawvere cylinder.

This is due to Cisinski (2006), §1.3.9.

By Prop. below, the Lawvere interval $\mathfrak{L}$ may be regarded as the universal cylinder object for Cisinski model structures on presheaf toposes.

## Properties

###### Proposition

The subobject classifier in any topos is an injective object, whence the Lawvere interval $L$ (Def. ) is a fibrant resolution of the terminal object in any Cisinski model structure on $Psh(A)$.

This is highlighted at the end of Cisinski (2006), §1.3.9 with reference (up to a typo) to MacLane Moerdijk (1992), IV §10.1:
###### Proof

Recall that for $L$ to be an injective object means that every solid span as below, where the vertical map is a monomorphism, admits a dashed lifting as shown:

Since in a Cisinski model structure, by definition, the monomorphisms are precisely the cofibrations, injective objects here are equivalently those for which the terminal map $L \to \ast$ is an acyclic cofibration.

Hence assuming the solid diagram above, we show the existence of $\underline{f}$:

Here $f \colon B \to L$ classifies a subobject of $B$, which we denote $C \hookrightarrow B$. The point now is that, with $B \hookrightarrow A$ being a monomorphism, the composite

$C \hookrightarrow B \hookrightarrow A$

exhibits $C$ also as a subobject of $A$, which as such is classified by some map $\underline{f} \colon A \to L$, and we claim that this serves as the desired extension. To see that indeed this $\underline{f}$ makes the above triangle commute, consider the following commuting diagram:

Here

• the right rectangle is the pullback square that witnesses $\underline{f}$ as the classifying map of $C \hookrightarrow A$, by definition,

• the left rectangle is the fiber product $B \cap_A C$ computed via the pasting law for pullbacks to be isomorphic to $C$, using that $C$ is already a subobject of $A$ (which gives the factorization in the middle) and then using (for the two squares on the left) that:

1. the fiber product of a monomorphism with itself is its domain (this Prop.),

2. isomorphisms are preserved by pullback (this Prop.).

This implies, again by the pasting law, that the total square is a pullback, hence that the total bottom map classifies $C$. But $C$ was defined to be classified by $f$, and so the uniqueness of subobject classifying maps implies that the total bottom map equals $f$, which was to be shown.

###### Proposition

Given any small set of monomorphisms in $Psh(A)$, there exists the smallest Cisinski model structure for which those monomorphisms are acyclic 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 $\mathbf{\Lambda}_\mathfrak{L}(S,M)$, we get exactly the contravariant model structure on the category of simplicial sets.