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smooth E-infinity-groupoid

Contents

under construction

Contents

Idea

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

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

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

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

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

HcoDiscΓDiscΠB. \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

Constructions

Multiplicative group

Write

𝔾 mB \mathbb{G}_m \in \mathbf{B}

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

𝔾 m:RR ×. \mathbb{G}_m \;\colon\; R \mapsto R^\times \,.

This is the canonical group object in B\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

𝔾 m= × \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 E_\infty-groupoid.

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

𝔾 m:(U,A)GL 1(A)C (U, ×), \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 AA with the abelian group of non-vanishing complex-valued smooth functions on XX.

Last revised on May 21, 2014 at 12:50:22. See the history of this page for a list of all contributions to it.