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lined topos

**differential geometry** **synthetic differential geometry** ## Axiomatics ## * smooth topos * infinitesimal space * amazing right adjoint * Kock-Lawvere axiom * integration axiom * microlinear space * synthetic differential supergeometry * super smooth topos * . infinitesimal cohesion * de Rham space * formally smooth morphism, formally unramified morphism, formally étale morphism * jet bundle ## Models ## * Models for Smooth Infinitesimal Analysis * Fermat theory * smooth algebra ($C^\infty$-ring) * smooth locus * smooth manifold, formal smooth manifold, derived smooth manifold * smooth space, diffeological space, Frölicher space * smooth natural numbers * Cahiers topos * smooth ∞-groupoid * synthetic differential ∞-groupoid ## Concepts * tangent bundle, * vector field, tangent Lie algebroid; * differentiation, chain rule * differential forms * differential equation, variational calculus * Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space * connection on a bundle, connection on an ∞-bundle * Riemannian manifold * isometry, Killing vector field ## Theorems * Hadamard lemma * Borel's theorem * Boman's theorem * Whitney extension theorem * Steenrod-Wockel approximation theorem * Poincare lemma * Stokes theorem * de Rham theorem * Chern-Weil theory ## Applications * Lie theory, ∞-Lie theory * Chern-Weil theory, ∞-Chern-Weil theory * gauge theory * ∞-Chern-Simons theory * Klein geometry, Klein 2-geometry, higher Klein geometry * Euclidean geometry, Cartan geometry, higher Cartan geometry * Riemannian geometry * gravity, supergravity

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Contents

Defnition

Definition

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

  • a ringed topos (𝒯,k)(\mathcal{T}, k)

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

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

Variations

Constructions in lined toposes

Path objects

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

As described there, this canonically induces

  • a cosimplicial object Δ R:Δ𝒯\Delta_R : \Delta \to \mathcal{T}

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

    that sends each object in the topos to a simplicial object

    XX Δ R X \mapsto X^{\Delta_R^\bullet}

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

Contractible objects

The following terminology is sometimes useful.

Terminology

(contractible object)

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

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

Examples

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

    R=(*01*). R = ({*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}) \,.
    Lemma

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

    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 VV \in CartSp every boundary of a simplex Δ[k]Π( n)(V)\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V) extends through Δ[k]Δ[k]\partial \Delta[k] \hookrightarrow \Delta[k]:

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

    Δ[n] ( n) Δ R (V) Δ[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 synth𝕃CartSp_{synth} \subset \mathbb{L} be the full subcategory of smooth loci on those smooth loci of the form n×D k(r)\mathbb{R}^n \times D_k(r), where D k(r)D_k(r) is the infinitesimal space of kkth order infinitesimal neighbours of the origin in r\mathbb{R}^r.

    The line object is again *01*{*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*} as in the above example. Crucially, the infinitesimal spaces D k(r)D_k(r) all have a unique point *D k(r){*} \to D_k(r). Accordingly, there is also a unique morphism R nD k(r)R^n \to D_k(r) for all nn. It follows that simplices in R n×D k(r)R^n \times D_k(r) are simplices in R nR^n as above, and trivial as maps to the D k(r)D_k(r)-factor. Hence the above argument carries over to this case and shows that all the n×D k(r)\mathbb{R}^n \times D_k(r) are contractible.

Revised on August 15, 2010 20:04:29 by Toby Bartels (75.88.65.54)