On the ordered set of real numbers, does sheaf+cosheaf imply constant? I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it.
Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers regarded as an ordered set), valued in the category of abelian groups.
Suppose that for every real number $s$, both of the maps
$$
P(s) \to \varprojlim_{t < s} P(t), \qquad P(s) \leftarrow \varinjlim_{t > s} P(t)
$$
are quasi-isomorphisms, where the lims indicate homotopy limit and colimit.  Does it follow that all of the maps $P(s_1) \to P(s_2)$ are quasi-isomorphisms?
When the $P(s)$ have homology groups that are uniformly bounded above, this is a result of Kashiwara.  It plays a key role in the foundations of microlocal sheaf theory, especially for sheaves that are not necessarily constructible.
Dima Tamarkin showed me how to remove the boundedness assumption, if you replace abelian groups by $k$-vector spaces for a field $k$. In that case the abelian category of presheaves of vector spaces on $\mathbf{R}$ has homological dimension $1$, so the derived category has a simple structure.
I'm interested in the case where $P$ takes values in the $\infty$-category of spectra, where you can formulate the same question, but already I don't know what to expect for abelian groups.
 A: Here is one possible way to proceed.
(Caveat lector: I haven't checked all the details carefully.)
Denote by S the sphere spectrum and by N the (contractible) spectrum that implements a nullhomotopy for S.
A map A→B of spectra is a weak equivalence if and only if
for any map Σ^k(S)→A with a nullhomotopy Σ^k(N)→B for the composition Σ^k(S)→A→B (i.e., Σ^k(S)→Σ^k(N)→B is Σ^k(S)→A→B),
the map Σ^k(S)→A is itself nullhomotopic via Σ^k(N)→A, with the composition Σ^k(N)→A→B being homotopic to the original map Σ^k(N)→B relative boundary Σ^k(S).
I refer to such data as a lift for a nullhomotopy.
(Depending on the (co)fibrancy of the spectra involved this criterion can be either made more strict or less strict.)
Both S and N are homotopy compact objects in spectra, hence their corepresentable functors commute with homotopy filtered colimits.
Suppose we want to prove P_s→P_t is a weak equivalence for s>t.
By the limit condition, a map Σ^k(S)→P_s defines maps Σ^k(S)→P_{s'} for all s>s'
and a nullhomotopy Σ^k(N)→P_t defines nullhomotopies Σ^k(N)→P_{t'} for all t>t'.
Denote by r the supremum of all r such that s≥r and there is a compatible system of lifts Σ^k(N)→P_q of the above nullhomotopy for all r>q≥t.
By the limit condition, the compatible system of lifts for all r>q yields a lift Σ^k(N)→P_r.
(Initially, we start with r=t and the lift is already given to us.)
Suppose s>r, then by the colimit condition the nullhomotopy lifts through some P_{r'}, s>r'>r,
which gives us a compatible system of lifts for r' by the limit condition, which contradicts the definition of r.
Thus s=r and we have constructed a lift Σ^k(N)→P_s as desired.
Addendum:
Here is how one can use model categories to simplify the above
discussion by eliminating homotopies.
Start with the model category of simplicial spectra or simplicial symmetric spectra.
Replace P fibrantly in the injective model structure on functors
R→Spectra; this guarantees that X_p is fibrant and X_p→X_q is a fibration for all p>q.
The weak equivalence criterion in the first paragraph can be formulated for fibrations between fibrant spectra by requiring a strict lift (in the ordinary sense) with respect to S→N.
The (cofiltered) homotopy limits can be computed as strict limits using fibrancy,
and the (filtered) homotopy colimits can be computed as strict colimits
by the compact generatedness of the model category of simplicial (symmetric) spectra.
With these conventions, the argument in the second paragraph can now be interpreted strictly.
