nLab smooth E-infinity-groupoid

under construction

Idea

For $\mathbf{B} = Sh_\infty(CRing_\infty^{op}, et)$ the (∞,1)-topos of E-∞ geometry, let

$$\mathbf{H} \coloneqq Sh_\infty(SmoothMfd, \mathbf{B})$$

be the (∞,1)-category of (∞,1)-sheaves on the site of smooth manifolds with values in $\mathbf{B}$.

An object in this $\mathbf{H}$ combines the properties of a smooth ∞-groupoid and an object in E-∞ geometry, hence might be called a “smooth $E_\infty$-groupoid”.

It is useful to regard this as a cohesive (∞,1)-topos over $\mathbf{B}$

$$\mathbf{H} \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}} \mathbf{B} \,.$$

As such this appears for instance in the discussion at

• differential algebraic K-theory

• equivariant elliptic cohomology.

Constructions

Multiplicative group

Write

$$\mathbb{G}_m \in \mathbf{B}$$

for the sheaf which sends each ring to its ∞-group of units

$$\mathbb{G}_m \;\colon\; R \mapsto R^\times \,.$$

This is the canonical group object in $\mathbf{B}$. The mapping stacks into it are the Picard ∞-stacks.

(…)

For E-∞ rings over the complex numbers, hence E-∞ algebras over $\mathbb{C}$, the multiplicative group

$$\mathbb{G}_m = \mathbb{C}^\times$$

naturally carries both the structure of an object in smooth ∞-groupoids and in E-∞ geometry, which may be combined to the structure of a smooth $E_\infty$-groupoid.

For $U \in SmthMfd$ and $A \in CRing_\infty(\mathbb{C})$ let $\mathbb{G}_m \in \mathbf{H}$ be given by

$$\mathbb{G}_m \;\colon\; (U,A) \mapsto GL_1(A)\otimes C^\infty(U,\mathbb{C}^\times) \,,$$

where on the right we have the ∞-groupoid underlying the abelian ∞-group which is the tensor product of the ∞-group of units of $A$ with the abelian group of non-vanishing complex-valued smooth functions on $X$.