Schreiber
∞-Lie theory

The higher categorical refinement of Lie theory we call $\mathrm{\infty }$-Lie theory . It studies

the interplay between

• ∞-Lie groupoids

and their infinitesimal version

• ∞-Lie algebroids.

We take the context in which $\mathrm{\infty }$-Lie theory is to be formulated to be an (∞,1)-topos $H_{\mathrm{synth}}$ that is modeled by a model category of simplicial sheaves $\mathrm{sSh}(C)$ such that the sheaf topos ${\mathrm{Spaces}}_{\mathrm{synth}}:=\mathrm{Sh}(C)$ is a model for synthetic differential geometry.

Conceptually this means that

The context is a smooth (∞,1)-topos $H_{\mathrm{synth}}$ of synthetically smooth ∞-groupoids.

Using the notion of infinitesimal objects in the synthetic differential geometry topos ${\mathrm{Spaces}}_{\mathrm{synth}}=\mathrm{Sh}(C)$ we obtain a notion of infinitesimal smooth $\mathrm{\infty }$-groupoids. These we identify as ∞-Lie algebroids.

An ∞-Lie algebroid is an infinitesimal ∞-Lie groupoid.

The archetypical example of an ∞-Lie algebroid is the infinitesimal version of the path ∞-groupoid $\Pi (X)$ of some Lie $\mathrm{\infty }$-groupoid $X$: the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$. For $X$ an ordinary manifold this is usually known as the tangent Lie algebroid $TX$ of $X$.

The – finite – differential adjunction involving the path ∞-groupoid

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

is thereby accompanied by its infinitesimal version, the infinitesimal differential adjunction involving the infinitesimal path ∞-groupoid

$${\Pi }^{\inf }(-):H_{\mathrm{synth}}\overleftarrow{\to }{H}_{\mathrm{synth}}:(-{)}_{\mathrm{flat}}^{\inf }.$$\Pi^{inf}(-) : \mathbf{H}_{synth} \stackrel{\leftarrow}{\to} \mathbf{H}_{synth} : (-)^{inf}_{flat} \,.

We conceive $\mathrm{\infty }$-Lie theory effectively as concerned with the composition of these two adjunctions

$$H_{\mathrm{synth}}\stackrel{\stackrel{\Pi (-)}{\leftarrow }}{\stackrel{(-{)}_{\mathrm{flat}}}{\to }}{H}_{\mathrm{synth}}\stackrel{\stackrel{(-{)}_{\mathrm{flat}}^{\inf }}{\leftarrow }}{\stackrel{{\Pi }^{\inf }(-)}{\to }}{H}_{\mathrm{synth}}.$$\mathbf{H}_{synth} \stackrel{\stackrel{\Pi(-)}{\leftarrow}}{\stackrel{(-)_{flat}}{\to}} \mathbf{H}_{synth} \stackrel{\stackrel{(-)^{inf}_{flat}}{\leftarrow}}{\stackrel{\Pi^{inf}(-)}{\to}} \mathbf{H}_{synth} \,.

This induces a notion of

• ∞-Lie differentiation and integration

  $\mathrm{Lie}(-):H_{\mathrm{synth}}\overleftarrow{\to }{H}_{\mathrm{synth}}:\exp (-)$

that revolves around the concept of

• ∞-Lie algebroid valued differential forms.