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Consider a diagram D: I→ChR of real connective chain complexes. In the example I have in mind all chain complexes are concentrated in some fixed degree n.

There is a canonical map lim D → holim D from the limit of D to the homotopy limit of D. I would like to have a practical criterion for determining when this map is a quasiisomorphism.

Of course, a sufficient condition for this would be the injective fibrancy of D. However, injective fibrancy seems to be difficult to check in general.

Thus I'm wondering if the fact that we are in the category of real chain complexes, which has rather nice properties (e.g., every epimorphism splits) might yield a more practical criterion.

Is there a practical way to check whether the canonical map lim D → holim D for a diagram D: I→ChR of real chain complexes is a quasiisomorphism?

Motivation for this question comes from a desire to compute the homotopy mapping space of simplicial presheaves on the site of smooth manifolds of the form Hom(F,BnR). Here Hom denotes the homotopy mapping space (i.e., the simplicial mapping space with the source cofibrantly replaced and the target fibrantly replaced), BnR denotes the Eilenberg—MacLane object corresponding to the representable presheaf R of real numbers (note that R here is not discrete), and F is the smooth singular set of some smooth manifold X, i.e., Fn(U) := {Δn × U → X}.

Although the simplicial presheaf BnR is fibrant in the local projective model structure, the simplicial presheaf F is not cofibrant, the primary obstacle being the fact that the corresponding simplicial components Fn are infinite-dimensional (they are the mapping spaces Map(Δn,X)), whereas cofibrancy in the projective structure requires them to be coproducts of retracts of representables, in particular they must be finite-dimensional.

One can try to circumvent this problem by observing first that any simplicial presheaf is equivalent to the homotopy colimit of its simplicial components Fn: Hom(F,BnR) = Hom(hocolimn Fn, BnR) = holimn Hom(Fn, BnR). Since R is contractible, it is plausible to expect that the higher cohomology of Fn with coefficients in R must vanish, in particular we hope to have Hom(Fn, BnR) = Bn Hom(Fn, R) = Bn C(Fn).

To calculate Hom(Fn,BnR) we can use the above trick one more time and observe that every presheaf of sets is equivalent to the homotopy colimit of its elements, hence we have Hom(Fn, BnR) = Hom(hocolimU→Fn U, BnR) = holimU→Fn Hom(U, BnR) = Γ holimU→Fn C(U)[n], the latter equality coming from the fact that U is representable and therefore cofibrant. (The functor Γ is the Dold—Kan functor that sends a chain complex to the corresponding simplicial set.) If the latter homotopy limit coincides with the corresponding limit, then we have holimU→Fn C(U)[n] = limU→Fn C(U)[n] = C(Fn)[n], which answers the original question.

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Since all your complexes are concentrated in a fixed degree $n$, the cohomology of the homotopy limit is $H^{n+r}(\operatorname{holim}D)=\lim^rD^n$, hence a criterion would be the vanishing of higher derived limits $\lim^rD^n=0$, $r>0$. This holds for all finite direct categories $I$, for instance, but in general it is complicated to know. I guess this doesn't help much.

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  • $\begingroup$ Do you know a reference for vanishing of higher homotopy limits for finite categories? $\endgroup$ – Dmitri Pavlov Nov 21 '13 at 10:29
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    $\begingroup$ Sorry, I was assuming that categories were direct, and in that case it's sort of trivial, otherwise it's blatantly false. I've edited my answer. $\endgroup$ – Fernando Muro Nov 21 '13 at 12:19

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