nLab Lawvere interval

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

The Lawvere cylinder is universal in the sense that any $L$-homotopy lifts to a homotopy for any other good cylinder object.

Proposition

Let $A$ be an object of a topos. Given a diagram

$$A \amalg A \stackrel{(\delta^0, \delta^1)}{\to} C \stackrel{\sigma}{\to} A$$

where the copies of $A$ are disjoint subobjects of $C$ splitting $\sigma$, then there is a morphism of cylinders $C \to L \times A$.

Proof

Let $\chi^i : C \to L$ be the characteristic function for $\delta^i$. That the copies of $A$ are disjoint subobjects implies $\chi^0 \leq \neg \chi^1$, and so $\chi^0 \vee \chi^1 \leq\chi^1 \vee \neg \chi^1$.

Since $x \mapsto x \vee \neg x$ is the characteristic function of $\lambda : 1 \amalg 1 \to L$, the morphism $(\chi^1, \sigma) : C \to L \times A$ pulls $A \times \lambda$ back to a subobject containing $\delta$. Compatibility with $\sigma$ is obvious, and so it is a morphism of cylinder objects.

Monomorphisms and trivial fibrations

In a topos, lifting problems against monomorphisms can be described in terms of partial maps, giving an elementary characterization of trivial fibrations:

Proposition

Let $p : X \to S$. Suppose the partial map classifier $X \hookrightarrow \mathrm{Opt}_S(X)$ exists in the slice category over $S$. Then $\mathrm{Opt}_S(X)$ represents the presheaf (in $A$) of diagrams of the form modulo the identification of equivalent subobjects $B \subseteq A$.

Corollary

If $p : X \to S$ is an arrow in a topos $\mathbf{E}$, it has the right lifting property against monomorphisms iff the restriction $X \times L \to \mathrm{Opt}_S(X)$ is a split epi.

Proof

$X \times L$ represents the presheaf (in $A$) of all such diagrams that have a filler $A \to X$, since $X \to A$ determines the lower triangle and from that, $X \to L$ determines the upper triangle.

Thus $p$ has the claimed property iff $X \to L \to \mathrm{Opt}_S(X)$ represents an epimorphism of presheaves, which is equivalent to being a split epi.

Proposition

If $p : X \to S$ is an arrow in a topos $\mathbf{E}$, it has the right lifting property against monomorphisms iff the insertion $X \hookrightarrow \mathrm{Opt}_S(X)$ has a retraction.

Proof

By the properties of partial map classifiers, every lifting problem has a unique factorization where the left square is a pullback. So $p$ has the right lifting property against the mono $X \to \mathrm{Opt}_S(X)$ iff it has the right lifting property against all monos.

Corollary

If $p : X \to S$ is an arrow in a topos $\mathbf{E}$, then $X \hookrightarrow \mathrm{Opt}_S(X) \to S$ is a factorization into a monomorphism followed by a morphism with the right lifting property against monomorphisms.

Proof

In any topos (in particular $\mathbf{E}_{/S}$), the both inclusions $\mathrm{Opt}(X) \hookrightarrow \mathrm{Opt}(\mathrm{Opt}(X))$ are split by the operation of taking the intersection of domains.

The argument spelled out in the case of $X = 1$ and $\mathrm{Opt}(X) = L$ is spelled out in

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.

Anodyne maps and fibrant objects

For the Lawvere interval in a presheaf topos, Cisinski’s anodyne completion operation stops at the first step $\Lambda_L(\varnothing, \mathrm{mono}) = \Lambda_L^0(\varnothing, \mathrm{mono})$ which is given by pushout products of monomorphisms with the endpoints $\mathbf{1} \to L$, due to the associativity and symmetry of the pushout product. This leads to

Definition

In a topos, say that an object is ($\mathfrak{L}$-)-fibrant iff both endpoint evaluations $X^L \to X$ are trivial fibrations.

Proposition

In a presheaf topos $\mathbf{E}$, the following are equivalent for an arrow $p : X \to S$:

1. $p$ is a fibrant object of $\mathbf{E}_{/S}$

2. $X^L \to S^L \times_S X$ is a trivial fibration, for both endpoints $\mathbf{1} \to L$.

3. $p$ is a naive $\mathfrak{L}$-fibration in the Cisinski model structure

4. $p$ has the right lifting property against $\mathfrak{L}$-anodyne maps in the Cisinski model structure

Proof

(3) and (4) restate the fact the anodyne maps and naive fibrations are constructed as a weak factorization system in Cisinski’s construction.

The pullback power of $p$ by $\mathbf{1} \to L$ has the right lifting property against a monomorphism $i$ iff $p$ has the right lifting property against the pushout product of $i$ with $\mathbf{1} \to L$, by Joyal-Tierney calculus. So (2) is equivalent to (3) and (4).

(1) is the assertion that $p \to 1_S$ is a naive $\mathfrak{L}_S$-fibration. The generating anodyne morphisms there are the pushout products $(A \subseteq B) \overline{\times_S} (S \to S \times L)$. But this is just the pushout product $(A \subseteq B) \overline{\times} (1 \to L)$ equippped with a map $B \to S$. So this is equivalent to $p$ being a naive $\mathfrak{L}$-fibration.

In a locally presentable topos, the small object argument is used to construct the (anodyne map, naive fibration) factorization system, and in particular to construct fibrant replacements.

The condition of being a a fibrant object is equivalent to asking $X^L \to \mathrm{Opt}_X(X^L)$ to have a retraction, or equvialently for a solution to the lifting problem

In a presheaf topos, the left vertical map will turn out to be a trivial cofibration. This gives a concrete construction for the small argument to be applied to in order to construct fibrant replacements.

References

• Denis-Charles Cisinski, Les préfaisceaux comme types d’homotopie, Astérisque 308 Soc. Math. France (2006), 392 pages [numdam:AST_2006__308__R1_0 pdf]