It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-Kan correspondence
to pass to double complexes and then applying
the totalization functor for double complexes,
possibly applying the truncation functor afterward.
For example, this is claimed (without proof)
in Dugger's notes on homotopy colimits, see Section 16.8
in http://math.uoregon.edu/~ddugger/hocolim.pdf.
In the above, “homotopy (co)limit” is used in the abstract
∞-categorical sense, i.e., the homotopy terminal (respectively
initial) object in the ∞-category of (co)cones.
It can be presented as the appropriately derived functor
of the ordinary (co)limit functor in the setting
of (stable) model categories or as the quasicategorical (co)limit
in the setting of stable quasicategories.
Is there a written proof of this result in the literature?
What about the case of unbounded chain complexes?
Best Answer
Rodríguez González, Beatriz(E-CSIC-IM) Simplicial descent categories. (English summary) J. Pure Appl. Algebra 216 (2012), no. 4, 775–788