More questions about Verdier duality (and related math) The first set of questions can be found here: Understanding (the wiki page on) Verdier duality
I'm fairly confident that I understand something wrong, so I'll write down here clearly what my set of beliefs is about what is right, and you feel free to shoot down any falsehood:
Let $X$ be our geometric object, be it a topological space, variety, scheme, or what have you. I will do two cases, one Poincare duality, and the other Serre duality.
Serre duality
I will assume $X$ is nice (a variety, projective, smooth,... of dimension $n$). Here I will look at the (abelian) category of coherent $O_X$-modules. In this case the "dualizing module" is $\omega_X[n]$. I take that to mean that $[\mathcal{E},\mathcal{F}]$ is dual to $[\mathcal{F},\mathcal{E}\otimes \omega_X[n]]$.
I also believe $H^k(X,\mathcal{F})\cong [O_X,\mathcal{F}[k]]$. Going from there, the rest is easy: $H^k(X,\mathcal{F})\cong [O_X,\mathcal{F}[k]]\cong [$dual of $\mathcal{F},O_X[k]]$, which is dual to $[O_X[k],\omega_X[n]\otimes$ dual of $\mathcal{F}]\cong [O_X, \omega_X \otimes$ dual of $\mathcal{F}[n-k]] \cong H^{n-k}(X,\omega_X \otimes $ dual of $\mathcal{F})$.
Great! However...
Poincare duality
Again assume $X$ is nice (orientable compact smooth manifold of dimension $n$). Let $K$ be field. Here I will look at the (abelian) category of $K$-vector spaces. Here the "dualizing sheaf" is $K[-n]$. I will continue somewhat similarly to the Serre duality case. I interpret the dualizing sheaf as meaning that $[\mathcal{E},\mathcal{F}]$ is dual to $[\mathcal{F},\mathcal{E}\otimes K[-n]]$ (here $\mathcal{E}$ and $\mathcal{F}$ are $K$-vector spaces).
For whatever reason, I believe that $H_k(X,\mathcal{E})\cong[\mathcal{E}[-k],D_X]$ ($D_X$ is the dualizing sheaf in general. But what does this even mean in general? For example in the Serre duality case, what would $H_k(X,\mathcal{F})$ even mean?). So:
$H_k(X,K) \cong [K[-k],K[-n]] \cong [K,K[k-n]] \cong H^{k-n}(X,K)$. Wait. What? Makes no sense!
If we use the "duality" it still makes no sense:
$[K,K[k-n]] \cong [K[k-n],K[-n]] \cong [K,K[-k]]\cong H^{-k}(X,K)$. What? Huh?
So in conclusion, I desperately want to have a handle on this, but I clearly don't. Hopefully just a nudge in the right direction would lead me to a better understanding of this yoga.
 A: I'm not sure if this will answer your question or not, but let $\mathbb{D}$ be the Verdier dualizing sheaf on the locally compact space $X$. If $M$ is a manifold, then $\mathbb{D}[-n]$, where $n$ is the dimension of $M$, is isomorphic in the derived category to the orientation sheaf $\omega$ on $M$ (see, for example, Borel's book on interesection cohomology, I think around section V.7). The Borel-Moore homology of $X$, which is equivalent to the ordinary homology of $X$ if $X$ is compact (otherwise it's the homology theory built from locally-finite chains) is defined to be $H^{BM}_{k}(X)=H^{-k}(X; \mathbb{D})$. If you unwind all that indexing business, it says that, for a compact manifold, $H_k(M)=H^{n-k}(M; \omega)$, which is probably the duality statement you're looking for. 
Of course showing that this has anything to do with classical Poincare duality (say via the cap product) is pretty far from obvious. This is part of the content of a paper I currently have in preparation with Jim McClure. 
