nLab lined topos

Contents

Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

Definition

A lined topos $(\mathcal{T}, R)$ is

• a ringed topos $(\mathcal{T}, k)$

(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.

Constructions in lined toposes

Path objects

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

As described there, this canonically induces

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

• a functor $\Pi : \mathcal{T} \to S \mathcal{T}$

that sends each object in the topos to a simplicial object

$X \mapsto X^{\Delta_R^\bullet}$

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

Contractible objects

The following terminology is sometimes useful.

Terminology

(contractible object)

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

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

Examples

• sheaves on topological spaces Let $Top'$ be a small version of the category of sufficiently nice topological spaces, for instance connected CW complexes. The canonical line object in $Sh(Top)$ is ${*} \stackrel{0}{\to} [0,1] \stackrel{1}{\leftarrow} {*}$ the standard topological interval. For $X \in Top$, $\Pi(X) = X^{\Delta_R^\bullet}$ 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 $\mathbb{R}^n$, for $n \in \mathbb{N}$. The canonical line object in $\mathcal{T} = Sh(CartSp)$ is the real line regarded as an interval object

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

In the lined topos $(\mathcal{T} = Sh(CartSp), R = \mathbb{R})$ the representable objects $\mathbb{R}^n$ 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 $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ extends through $\partial \Delta[k] \hookrightarrow \Delta[k]$:

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

$\array{ \partial \Delta[n] &\to& (\mathbb{R}^n)^{\Delta_R^\bullet}(V) \\ \downarrow & \nearrow \\ \Delta[n] } \,.$
• sheaves on cartesian smooth loci A small variation of the above example leads to smooth toposes with contractible representables:

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

The line object is again ${*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}$ 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 $\mathbb{R}^n \times D_k(r)$ are contractible.

Last revised on April 16, 2017 at 04:38:58. See the history of this page for a list of all contributions to it.