The objects in the smooth (∞,1)-topos are ∞-Lie groupoids that, as ∞-groupoids are of the most general kind.
Many ∞-Lie groupoids appearing in practice are (equivalent) to objects in sub--categories of much stricter -Lie groupoids. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general ∞-Lie groupoids. Therefore it is of interest to have various notions of strict ∞-Lie groupoids inside all of them.
One well-known such notion is given by the Dold-Kan correspondence. This identifies chain complexes of abelian groups with strict and strictly symmetric monoidal -groupoids.
Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:
the identification of crossed complexes of groupoids as precisely the strict ∞-groupoids. This has been studied in particular in nonabelian algebraic topology.
Here we develop some useful facts and tools for working with ∞-Lie groupoids that happen to be strict ∞-Lie groupoids.
There are two main ingredients:
restriction to abelian sheaf cohomology – Using the fact that the objects of the smooth (∞,1)-topos are modeled by simplicial sheaves symmetric monoidal -Lie groupoids are identified under the Dold-Kan correspondence with -graded chain complexes of sheaves. On these the general theory of nonabelian cohomology restricts to the familiar abelian sheaf cohomology. Standard tools from abelian sheaf cohomology are embedded this way into the general theory.
descent for strict -groupoid valued sheaves – There is a good theory descent for (presheaves) with values in strict -groupoids (more restrictive than the fully general theory but more general than abelian sheaf cohomology). This goes back to Ross Street. Its relation to the full theory has been clarified by Dominic Verity. This is described at Verity on descent for strict omega-groupoid valued presheaves.