An integration of ∞-Lie algebroid valued differential forms is an extension from the infinitesimal path ∞-groupoid? to the finite path ∞-groupoid
after injecting the ∞-Lie algebroid into one of its corresponding ∞-Lie groupoids .
Assume a context given by a smooth (∞,1)-topos with its notion of ∞-Lie theory.
For some ∞-Lie groupoid and a test domain, a morphism from the infinitesimal path ∞-groupoid? factors – by definition of ∞-Lie algebroid – through the ∞-Lie algebroid of
The first morphism here constitutes the set of ∞-Lie algebroid valued differential forms on that is encoded by .
We want to specify what it means to integrate these forms.
Whereas colors infinitesimal intervals on by infinitesimal morphisms in , its integration should be a rule that labels finite intervals by finite morphisms, such that the restriction to any infinitesimal interval within a finite interval reproduces the assignment of .
Formally, this desideratum clearly means that integration of is an extension
of through the inclusion of the infinitesimal path ∞-groupoid? in the path ∞-groupoid.
More generally, we can ask for a relative integration : we may have a situation where a coloring of finite intervals
is already given, but infinitesimally there is also a lift specified of this through some map , i.e. a diagram
Integration should exist when the map in principle admits such a lift and should be given by that lift. It should be unique up to equivalence.
We discuss a case in which integration of -Lie algebroid valued forms always exsist.
Let be an ∞-Lie algebroid.
Write for short in the following for the image of under the right adjoint of the infinitesimal path ∞-groupoid? functor . And write for the corresponding presheaf of 0-cells.
By the reasoning an ∞-Lie differentiation and integration we have
For simplicially discrete, every morphism
corresponding by adjunction to a morphism
induces a morphism
that fits into the diagram