Schreiber
∞-Lie groupoid

For our purposes here

$\mathrm{\infty }$-Lie groupoid is an ∞-groupoid that is equipped with a smooth structure, in generalization to how a Lie group is a group with smooth structure.

Definition

An $\mathrm{\infty }$-Lie groupoid as an object in a smooth (∞,1)-topos.

$\mathrm{\infty }$-Lie groupoids are the objects under discssion in ∞-Lie theory.

Remark The literature tends to reserve “Lie-groupoid” more restrictively for groupoids internal to Diff. But from the axiomatic point of view of synethetic differential geometry the objects in $H$ perfectly qualify as a good generalization of the notion of Lie group. It is of course true that $H$ contains objects that from a classical perspective appear “pathological”. But following the credo that a nice category with general objects is better than a badly-bebaved category of very specifi objects we take here the point of view that $H$ is the right context for the topic that should be called ∞-Lie theory and will use specific terms for specific objects in $H$. In particual we shall talk about geometric $\mathrm{\infty }$-Lie groupoids (…).

special cases

$\mathrm{\infty }$-Lie algebroids

An $\mathrm{\infty }$-Lie groupoid all of whose k-morphisms are infinitesimal is an ∞-Lie algebroid.

Every $\mathrm{\infty }$-Lie groupoid $A$ is approximated by its ∞-Lie algebroid $𝔞=\mathrm{Lie}(A)$. This is described at ∞-Lie differentiation and integration.

strict $\mathrm{\infty }$-Lie groupoids

For computations, an important sub-class of $\mathrm{\infty }$-Lie groupoids are the strict ∞-Lie groupoids. Under the Dold-Kan correspondence and its nonabelian generalizations, strict $\mathrm{\infty }$-Lie groupoids are the objects in smooth homological algebra and smooth nonabelian algebraic topology.

Kan-fibrant replacements of simplicial objects

When the ambient smooth (∞,1)-topos is presented by a model structure on simplicial sheaves then every simplicial sheaf in this context presents an $\mathrm{\infty }$-Lie groupoid: the one that is obtained from the simplicial sheaf by its objectwise Kan fibrant replacement.