Schreiber
smooth (∞,1)-topos

The notion of smooth (,1)(\infinity,1)-topos is an (∞,1)-topos version of that of smooth topos:

Contents

Idea

The archetypical (∞,1)-topos is ∞-Grpd, the (∞,1)-category of ∞-groupoids.

If we think of this as an (,1)(\infty,1)-Grothendieck topos it is that of (∞,1)-sheaves on the point:

-GrpdSh (*). \infty\text{-}Grpd \simeq Sh_{\infty}({*}) \,.

Following the logic of space and quantity this may be understood as saying that a bare ∞-groupoid without further structure gives just a prescription for how to map the point into it: there is an \infty-groupoid Hom(*,A)Hom({*},A) of ways of mapping the point into the \infty-groupoid AA, and that reproduces AA.

A Lie ∞-groupoid – or ∞-Lie groupoid as we shall say – should instead be an \infty-groupoid that comes with the additional information on what a (contractible) smooth family of points inside it should be. Accordingly, it should provide a rule that assigns to each (contractible) smooth family UU of points an \infty-groupoid Hom(U,A)Hom(U,A) of smooth maps of UU into AA.

This means that for a suitable site of smooth test spaces, an ∞-Lie groupoid should be an object in an (∞,1)-topos of (∞,1)-sheaves on CC

-LieGrpdSh (C). \infty\text{-}LieGrpd \simeq Sh_{\infty}(C) \,.

Under a smooth test space we shall understand an object in a site that models synthetic differential geometry.

Remarks

Lie terminology

Notice that an \infty-groupoid that may be probed by contractible ordinary manifolds is slightly more general than being an \infty-groupoid internal to diffeological spaces. Therefore what we call \infty-Lie groupoids here are considerably more general than some notion of \infty groupoids internal to manifolds. We shall still just say \infty-Lie groupoid for our definition, for brevity.

Our category of \infty-Lie groupoids is a nice category that contains some pathological objects:

it

  • supports a good general ∞-Lie theory

  • while restriction to special nice objects is a matter of concrete applications.

Smooth cohomology and differential refinement

The cohomology theory of the smooth (,1)(\infty,1)-topos H\mathbf{H} is smooth cohomology.

To refine this to differential cohomology we refine H\mathbf{H} to a structured (∞,1)-topos using the path ∞-groupoid.

Definition

Definition

(smooth (,1)(\infty,1)-topos)

Let CC be a site of smooth loci such that the category of sheaves Sh(C)Sh(C) equipped with the canonical line object R=C ()R = \ell C^\infty(\mathbb{R}) is a smooth topos.

Let then SPSh(C) loc projSPSh(C)_{loc}^{proj} and SSh(C) loc projSSh(C)_{loc}^{proj} be the local projective model structure on simplicial presheaves and Quillen equivalently the local projective model structure on simplicial sheaves on CC and let

H smoothSh (C)(SPSh(C) loc) (SSh(C) loc) \mathbf{H}_{smooth} \coloneqq Sh_\infty(C) \coloneqq (SPSh(C)_{loc})^\circ \simeq (SSh(C)_{loc})^\circ

be the (∞,1)-category presented by that. Then we call H\mathbf{H} a smooth (,1)(\infty,1)-topos.

Remarks
  1. The restriction that CC be a site of smooth loci is to ensure that there is a good notion of infinitesimal path ∞-groupoid? in H\mathbf{H}. But all the common Models for Smooth Infinitesimal Analysis are of this form.

  2. In practice we usually use smooth (,1)(\infty,1)-toposes whose underlying smooth topos has, as a lined topos, contractible representables.

    For this case the path ∞-groupoid functor extends (as discussed there) to a Quillen adjunction

    Π:SPSh(C) proj locSPSh(C) proj loc:() flat \Pi : SPSh(C)_{proj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{proj}^{loc} : (-)_{flat}

    and hence to an ∞-functor

    Π:HH:() flat \Pi : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat}

    on the (∞,1)-topos.

Revised on December 29, 2009 18:12:51 by Urs Schreiber (212.23.128.235)