Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives) I'm trying to understand the proof of Theorem 4.1 in the paper Multiple Polylogarithms and Mixed Tate Motives by AB Goncharov (http://arxiv.org/pdf/math/0103059v4.pdf). In it, the author uses cosheaf homology.
As far as I can tell, the global sections functor for cosheaves is right exact, so homology should be given by the left derived functors of the global sections functor. Similarly, the higher direct image functors should be the left derived functors of the standard direct image. The Grothendieck spectral sequence should be a homological spectral sequence. 
I have four questions:


*

*Is this correct?

*I understand that (see e.g. Hartshorne, Chapter III, Proposition 8.1) the cosheaf sending an open set $U$ to $H_q(p^{-1}(U))$ should be the $q$th direct image of the constant sheaf $\mathbb Z$. However, when on p.46 we are defining the cosheaf $\mathcal{R}_c$ and we are taking relative homology, what cosheaf are we taking the direct image of?

*Where does the exact sequence (110) come from? Do we always get an exact sequence from the joining of two complexes?

*Between (109) and (110), I assume that $R_i$ means left derived functor, since, as I mentioned, higher direct images of cosheaves are left, not right, derived functors. But what on Earth does he mean by the higher direct image of a subvariety (or complex of subvarieties)?
I'm guessing that the $q$th direct image of the complex of varieties should be interpreted as the cosheaf corresponding to the homology relative to the union of the varieties associated to that complex? Assuming that's the case, I'm a little unsure how to deal with the higher direct images of the truncation (maybe it corresponds to the hyperhomology of the complex on the inverse image of $U$? Then the special case of the whole complex makes sense since the hyperhomology of that complex is the relative homology. But if that's so, I don't know what to make of the hyperhomology of the truncation...)
 A: Let me begin with a reference: Marta Bunge and Jonathon Funk, Singular coverings of toposes. What follows is a very concise digest of very few of the features of that book.
Seems like the correct way of dualizing the correspondence between sheaves and local homeomorphisms is the following.
A continuous map $f:E\to X$ is a local homeomorphism iff the canonical map
$$
\operatorname{stalk}_x(f):=\varinjlim_{U\ni x}\Gamma\left(\left.f\right|_U\right)\to f^{-1}(x)
$$
given by
$$
(\text{germ at $x$ of a section $\sigma$})\mapsto\sigma(x)
$$
is bijective for any $x\in X$, where $\Gamma\left(\left.f\right|_U\right)$ denotes the set of all global sections of the map $\left.f\right|_U:f^{-1}(U)\to U$.
Now considering that $\Gamma$ is right adjoint to the inverse image, it is most natural to try the left adjoint to the inverse image. It does not always exist but in good cases (related to local connectedness) is given by connected components. So in case there are appropriate adjunctions $\pi_0\dashv f^*\dashv\Gamma$, it makes sense to consider the following condition on $f$ as a first approximation to the dual notion: it is a "co-local homeomorphism" if the canonical maps
$$
f^{-1}(x)\to\varprojlim_{U\ni x}\pi_0(f^{-1}(U))=:\operatorname{costalk}_x(f)
$$
given by
$$
f^{-1}(x)\ni e\mapsto\left(\text{connected component of $e$ in $f^{-1}(U)$}\right)_{U\ni x}
$$
are bijective. In this context then, it is natural to call matching families of connected components of inverse images of neighborhoods of a point cogerms (of cosections?) at that point. So every point in the inverse image of $x$ determines a cogerm at $x$, and the condition is that every cogerm is of this kind for a unique point in the inverse image.
This approach shows that cosheaves are "more special" than sheaves: to begin with, the $\pi_0$'s involved might be quite nasty. Even if they are nice, usually they are non-discrete. And even if they are discrete, still in general the inverse limit  carries a natural nondiscrete inverse limit topology, so it is natural to require of the above map to be not just a bijection but a homeomorphism. In this way one more or less gets complete spreads introduced by R. H. Fox in 1957 and studied by several people ever since. Their total spaces behave much better than those of local homeomorphisms (which might be non-Hausdorff even over very nice spaces). More or less, complete spreads are branched coverings.
There are several other subtleties to take into account, but, at any rate, Bunge and Funk build a theory dual to that of sheaves based on the notion of complete spread, and obtain a duality with several nice properties. In particular they deepen further the insight of of Lawvere that sheaves are "like functions" while cosheaves are "like distributions".
A: Cosheaves are indeed mysterious gadgets. On the one hand, cosheaves are everywhere, but on the other hand, someone used to thinking sheaf-theoretically may have some problems. I am very close to finishing an exposition on cosheaves, but need another week or so to put it on the arxiv. Bredon's book on sheaf theory has the most complete reference on cosheaves, so you might look there if you like.
AS you may know, pre-cosheaves are just covariant functors $\hat{F}:\mathrm{Open}(X)\to\mathcal{D}$ where $\mathcal{D}$ is some "data category" like Vect, Ab, or what have you. Cosheaves send covers (closed under intersection) to colimits and different covers of the same open set get sent to isomorphic colimits. The Mayer-Vietoris axiom is a good way of thinking about cosheaves and since homology commutes with direct limits, one can see that $H_0(-,k)$ is always a cosheaf. In particular, $H_0(-,\mathcal{L})$ is a cosheaf whenever $\mathcal{L}$ is a local system.
As you observed, since cosheaves are fundamentally colimit-y, they have left-derived functors rather than right-derived ones. Thus the answer to (1) is yes.
In regards to (2), one must be careful. I believe the answer is yes, but allow me to pontificate on the problem.
Filtered limits and finite colimits do not commute in most categories like Ab, Vect, or Set. This has serious ramifications through the theory of cosheaves. 
For example, it is not necessarily true that a sequence of cosheaves is exact iff it is exact on costalks. Here costalks are defined using (filtered) inverse limits rather than direct ones. 
Another very serious consequence is that Grothendieck's sheafification procedure cannot be dualized to give cosheafification. Thus the usual phrase 
"let blah by the cosheaf associated to the pre-cosheaf blah" 
is not necessarily well-founded because it is unclear how to cosheafify! People have solved this problem in the past by working with pro-objects (which corrects for this "filtered limits not commuting with finite colimits" asymmetry) and then they use Grothendieck's construction. However, for abstract categorical reasons one can check that cosheafification does exist for data categories like Vect (i have worked out a proof and haven't found in the literature anyone who claims to have proved this), we just don't have an explicit construction. That said, the usual description of the left-derived functor of the push-forward should still hold. 
On the other hand, if one works in the constructible setting, one can get the statements you would like. In particular, it is true that cosheaves constructible with respect to a cell structure are derived equivalent to sheaves constructible with respect to the same cell structure. I discovered independently my own proof, only to find that at least two other people have proved this before. However, in my opinion, the equivalence is the "correct" form of Verdier duality. A larger and updated exposition should be available soon.
