nLab
lined topos

Defnition

Definition

A lined topos $(𝒯,R)$ is

• a ringed topos $(𝒯,k)$

  (usually with the internal ring object $(k,+,\cdot )$ assumed to be commutative)

• and equipped with a choice $(R,+,\cdot )$ of internal commutative algebra object $(R,+,\cdot )$ over $k$ – the line object.

Variations

• A smooth topos is a lined topos where the line is required to be a smooth differentiable space with infinitesimal subspaces in a certain way. This is the basic type of object studied in synthetic differential geometry.

• A super smooth topos is a lined topos that is a smooth topos and in which the $k$-algebra structure on $R$ is refined to that of a $k$-superalgebra.

Constructions in lined toposes

Path objects

The line object $R$ in a lined topos $𝒯$ canonically has the structure of a cartesian interval object.

As described there, this canonically induces

• a cosimplicial object ${\Delta }_R:\Delta \to 𝒯$

• a functor $\Pi :𝒯\to S𝒯$

  that sends each object in the topos to a simplicial object

  $$X\mapsto X^{{\Delta }_R^{•}}$$X \mapsto X^{\Delta_R^\bullet}

  ( which may be interpreted as presenting the path ∞-groupoid of $X$).

Contractible objects

The following terminology is sometimes useful.

Terminology

(contractible object)

Let $(𝒯=\mathrm{Sh}(C),R)$ be a lined Grothendieck topos with respect to a site $C$.

Call an object $X\in 𝒯$ contractible with respect to the interval object $R$, if the simplicial sheaf $\Pi (X)=X^{{\Delta }_R^{•}}:C^{\mathrm{op}}\to$ SSet sends each object to a contractible simplicial set.

Examples

• sheaves on topological spaces Let $\mathrm{Top}\prime$ be a small version of the category of sufficiently nice topological spaces, for instance connected CW complexes. The canonical line object in $\mathrm{Sh}(\mathrm{Top})$ is $*\stackrel{0}{\to }[0,1]\stackrel{1}{\leftarrow }*$ the standard topological interval. For $X\in \mathrm{Top}$, $\Pi (X)=X^{{\Delta }_R^{•}}$ is the singular simplicial complex of $X$. This is contractible in the above sense precisely if $X$ is a contractible space in the standard sense.

• sheaves on cartesian spaces Let CartSp be the full subcategory of Diff on smooth manifolds of the form ${ℝ}^n$, for $n\in \mathbb{N}$. The canonical line object in $𝒯=\mathrm{Sh}(\mathrm{CartSp})$ is the real line regarded as an interval object

  $$R=(*\stackrel{0}{\to }ℝ\stackrel{1}{\leftarrow }*).$$R = ({*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}) \,.

  Lemma

  In the lined topos $(𝒯=\mathrm{Sh}(\mathrm{CartSp}),R=ℝ)$ the representable objects ${ℝ}^n$ are contractible with respect to $R$.

  Proof

  This is not quite as entirely trivial as it may seem on first sight, but follows directly from the Tietze extension theorem for smooth manifolds:

  we check that for all $V\in$ CartSp every boundary of a simplex $\partial \Delta [k]\to \Pi ({ℝ}^n)(V)$ extends through $\partial \Delta [k]\hookrightarrow \Delta [k]$:

  by the construction of the cosimplicial object ${\Delta }_R:\Delta \to \mathrm{Sh}(\mathrm{CartSp})$ we have that morphisms $\partial \Delta [k]\to \Pi ({ℝ}^n)(V)$ correspond to smooth maps from the boundary of a $V$-cylinder over the standard $k$-simplex in ${ℝ}^k\times V\to {ℝ}^n$. Since this is a closed subset of ${ℝ}^k\times V$, by the Tietze extension theorem these maps extend (apply the theorem to each of their components) to all of ${ℝ}^k\times V$, hence in particular to the standard $k$-simplex inside ${ℝ}^k$ defined by the interval object. This constitutes the required extension to a $V$-family of $k$-simplices in ${ℝ}^n$

  $$\begin{array}{ccc}\partial \Delta [n]& \to & ({ℝ}^n{)}^{{\Delta }_R^{•}}(V)\\ \downarrow & \nearrow \\ \Delta [n]\end{array}.$$\array{ \partial \Delta[n] &\to& (\mathbb{R}^n)^{\Delta_R^\bullet}(V) \\ \downarrow & \nearrow \\ \Delta[n] } \,.

• sheaves on cartesian smooth loci A small variation of the above example leads to smooth toposes with contractible representables:

  let ${\mathrm{CartSp}}_{\mathrm{synth}}\subset 𝕃$ be the full subcategory of smooth loci on those smooth loci of the form ${ℝ}^n\times D_k(r)$, where $D_k(r)$ is the infinitesimal space of $k$th order infinitesimal neighbours of the origin in ${ℝ}^r$.

  The line object is again $*\stackrel{0}{\to }ℝ\stackrel{1}{\leftarrow }*$ as in the above example. Crucially, the infinitesimal spaces $D_k(r)$ all have a unique point $*\to D_k(r)$. Accordingly, there is also a unique morphism $R^n\to D_k(r)$ for all $n$. It follows that simplices in $R^n\times D_k(r)$ are simplices in $R^n$ as above, and trivial as maps to the $D_k(r)$-factor. Hence the above argument carries over to this case and shows that all the ${ℝ}^n\times D_k(r)$ are contractible.