See for instance at Hurewicz theorem
Let be a commutative ring. Write
for the (∞,1)-category of (∞,1)-modules over .
There is an equivalence of (∞,1)-categories
This is the statement of the stable Dold-Kan correspondence, see at (∞,1)-category of (∞,1)-modules – Properties – Stable Dold-Kan correspondence.
between flat (∞,1)-module bundles/local systems and L-∞ algebroid representations of the tangent Lie algebroid of . From right to left the equivalece is established by sending an L-∞ algebroid representation given (as discussed there) by a flat -graded connection on bundles of chain complexes (via prop. 1), to its higher holonomy defined in terms of iterated integrals.
This is the main theorem in (Block-Smith 09).
for the (∞,1)-colimit functor.
We may think of equivalently as
This is a classical basic (maybe folklore) statement. Here is one way to see it in full detail.
First notice that the (∞,1)-colimit of functors out of ∞-groupoids and constant on the tensor unit in is by definition the (∞,1)-tensoring operation of over ∞Grpd. Now if we find a presentation of by a simplicial model category the by the dicussion at (∞,1)-colomit – Tensoring and cotensoring – Models this (∞,1)-tensoring is given by the left derived functor of the sSet-tensoring in that simplicial model category.
To obtain this, use prop. 1 and then the discussion at model structure on chain complexes in the section Projective model structure on unbounded chain complexes which says that there is a simplicial model category structure on the category of simplicial objects in the category of unbounded chain complexes which models , and whose weak equivalences are those morphisms that produce quasi-isomorphism under the total chain complex functor.
where on the right we have the coend over the simplex category of the tensoring (of simplicial sets with simplicial objects in the category of unbounded chain complexes) of the standard cosimplicial simplex with the simplicial diagram constant on the tensor unit chain complex.
See also at Dold-Thom theorem.