Schreiber
smooth (∞,1)-topos

The notion of smooth $(\mathrm{\infty }\mathrm{,1})$-topos is an (∞,1)-topos version of that of smooth topos:

• where a smooth topos is a topos of generalized smooth spaces, that admits a notion of smooth infinitesimal spaces;

• a smooth $(\mathrm{\infty }\mathrm{,1})$-topos is an (∞,1)-topos of ∞-Lie groupoids that admits a notion of infinitesimal $\mathrm{\infty }$-Lie groupoids : of ∞-Lie algebroids.

Idea

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

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

$$\mathrm{\infty }\text{-}\mathrm{Grpd}\simeq {\mathrm{Sh}}_{\mathrm{\infty }}(*).$$\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 $\mathrm{\infty }$-groupoid $\mathrm{Hom}(*,A)$ of ways of mapping the point into the $\mathrm{\infty }$-groupoid $A$, and that reproduces $A$.

A Lie ∞-groupoid – or ∞-Lie groupoid as we shall say – should instead be an $\mathrm{\infty }$-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 $U$ of points an $\mathrm{\infty }$-groupoid $\mathrm{Hom}(U,A)$ of smooth maps of $U$ into $A$.

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 $C$

$$\mathrm{\infty }\text{-}\mathrm{LieGrpd}\simeq {\mathrm{Sh}}_{\mathrm{\infty }}(C).$$\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 $\mathrm{\infty }$-groupoid that may be probed by contractible ordinary manifolds is slightly more general than being an $\mathrm{\infty }$-groupoid internal to diffeological spaces. Therefore what we call $\mathrm{\infty }$-Lie groupoids here are considerably more general than some notion of $\mathrm{\infty }$ groupoids internal to manifolds. We shall still just say $\mathrm{\infty }$-Lie groupoid for our definition, for brevity.

Our category of $\mathrm{\infty }$-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 $(\mathrm{\infty }\mathrm{,1})$-topos $H$ is smooth cohomology.

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

Definition

Definition

(smooth $(\mathrm{\infty }\mathrm{,1})$-topos)

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

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

$$H_{\mathrm{smooth}}≔{\mathrm{Sh}}_{\mathrm{\infty }}(C)≔(\mathrm{SPSh}(C{)}_{\mathrm{loc}}{)}^{\circ }\simeq (\mathrm{SSh}(C{)}_{\mathrm{loc}}{)}^{\circ }$$\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 $H$ a smooth $(\mathrm{\infty }\mathrm{,1})$-topos.

Remarks

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

2. In practice we usually use smooth $(\mathrm{\infty }\mathrm{,1})$-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

   $$\Pi :\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}:(-{)}_{\mathrm{flat}}$$\Pi : SPSh(C)_{proj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{proj}^{loc} : (-)_{flat}

   and hence to an ∞-functor

   $$\Pi :H\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}$$\Pi : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat}

   on the (∞,1)-topos.