Understanding (the wiki page on) Verdier duality My familiarity with concepts related to derived categories is only tangential, and little by little I intend to get more comfortable with them. I was playing around with Caldararu's introduction to the topic, and looking up various things on the web.
Here is my question (that I have every confidence is trivial for experts):
On the wiki page on Verdier duality http://en.wikipedia.org/wiki/Verdier_duality it says the following: Let $F$ be a field, and $X$ a finite dimensional (dimension is defined here cohomologically, but for our purposes a finite dimensional manifold will do) locally compact space.
In the part about Poincare duality, it says:
$H^k(X,F)=[F,X[k]]$.
What is the interpretation of this notation? As I see it, $[F,X[k]]$ means $Hom(F,X[k])$ in the derived category. But this means that $X$ is seen as a complex. How? And why would $Hom(F,X[k])$ equal $H^k(X,F)$?
 A: I think here is how one should understand the last paragraph of the wiki. 
Consider $f:X\to pt$. We have (all functors are derived and my $Hom$ are sheaf $Hom$)
$$
Hom_{pt}(f_!F,F)=f_*Hom_X(F,f^!F)
$$
$f_!F=\Gamma_c(X,F)$, so the l.h.s. computes the dual of the cohomology with compact support of the constant sheaf $F$, i.e. the dual of the cohomology with compact support of $X$. 
$Hom_X(F,f^!F)= \Gamma(f^!F)$ and thus the r.h.s. is $f_*(\Gamma(f^!F))=\Gamma(X,f^!F)$, the homology of $X$. 
EDIT : to answer precisely the question I include a summary of the comments. 


*

*there is a typo in the wiki: $[F,X[k]]$ should be understood as $[F,F[k]]$.

*for any sheaf of $F$-modules $S$ (concentrated in degree 0), $[F[−k],S]=H^k(X,S)$. 

*Contrary to what is claimed in the wiki, there is no duality between $H^k(X,F)=[F[−k],F]$ and $H_k(X,F)=[F[−k],D_X]$ (where $D_X:=f^!F$ for $f:X\to pt$). The duality is, either between $H^k_c(X,F)$ and $H_k(X,F)$, or between $H^k(X,F)$ and $H_k^{BM}(X,F)$. And as far as I understand $H^*(X,S)$ is dual to $H_c^*(X,S^\vee)$. 
A: I just revamped what was written. Perhaps now it is more understandable:New Wiki Entry on Verdier duality.
