Schreiber smooth (∞,1)-topos

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

Idea

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

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

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

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 $Hom({*},A)$ of ways of mapping the point into the $\infty$-groupoid $A$, and that reproduces $A$.

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 $U$ of points an $\infty$-groupoid $Hom(U,A)$ 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 $C$

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

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 $(\infty,1)$-topos $\mathbf{H}$ is smooth cohomology.

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

Definition

Definition

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

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

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

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

be the (∞,1)-category presented by that. Then we call $\mathbf{H}$ a smooth $(\infty,1)$-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 $\mathbf{H}$. But all the common Models for Smooth Infinitesimal Analysis are of this form.

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

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

   and hence to an ∞-functor

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

   on the (∞,1)-topos.