I am thinking about orientation of a connected manifold $M$ of dim $n$ as a sheaf.
There are two definitions I could use, the first is the sheaf associated to the presheaf $$U\mapsto H_n(M,M-U;R).$$ The second is the sheaf of sections of generators of the fibration $R^*\to \tilde{M}_R\to M$, where $R$ is a ring and $R^*$ is the discrete group of units of $R$ and $\tilde{M}_R$ is the $R$-orientable cover of $M$.
I have the following questions:
1. Are the two definitions the same?
2. Theorem 3.26 of Hatcher seems to translate to that if $M$ is closed, then it is orientable iff the orientation sheaf has a global section that generate stalk-wise, i.e. it is a principal $\underline{R}$-module.
3. Lemma 3.27 seems to be saying if $M$ is closed, then the presheaf above is already a sheaf.
Are these correct? Feel free to tell me more things or give me references.
(1) Your two definitions can't be the same because one of them restricts to units in $R$ and the other does not. To be more precise, let $F_0$ be the presheaf of $R$-modules $U\mapsto H_n(M,M-U;R)$ and let $F$ be its sheafification. Let $G$ be the sheaf of continuous sections of the bundle of $R$-modules on $M$ whose fiber at $x$ is $H_n(M,M-\{x\};R)$, and let $G^*\subset G$ be the subsheaf of sections which generate every fiber. Then $F$ is your first sheaf, and $G^*$ is your second sheaf. But these are not isomorphic; rather, $F\cong G$. To get this isomorphism, note that there is a canonical map $F_0\to G$ (given an element of $H_n(M,M-U;R)$, restrict it to $H_n(M,M-\{x\};R)$ for each $x\in U$), and this map induces an isomorphism on stalks (since any point has arbitrarily small neighborhoods on which both $F_0$ and $G$ evaluate to $R$, with the map being the identity). This map thus induces an isomorphism after sheafifying, giving an isomorphism $F\to G$. It is $G^*$ which is normally referred to as the "sheaf of $R$-orientations", not $F$. If you like, you can identify $G^*$ as a subsheaf $F^*$ of $F$ via the isomorphism; it can be described as the subsheaf of sections which generate every stalk as an $R$-module (or, more elegantly, as sheaf of isomorphisms of $\underline{R}$-modules $\operatorname{Iso}_{\underline{R}}(\underline{R},F)$). While $F\cong G$ canonically has the structure of a sheaf of $R$-modules, $F^*\cong G^*$ is merely a sheaf of $R^*$-sets.
(2) Theorem 2.36(a) says that if $M$ is closed, connected, and $R$-orientable, then there is a global section of $F_0$ that makes $F$ a principal $\underline{R}$-module, i.e. it is (noncanonically) isomorphic to the constant sheaf $\underline{R}$ as a sheaf of $R$-modules. Note that Hatcher's definition of "$R$-orientable" is exactly that $G$ is a principal $\underline{R}$-module (or equivalently, that $G^*\cong\operatorname{Iso}_{\underline{R}}(\underline{R},G)$ has a global section), so it is trivial that in that case $F\cong G$ is also principal. The nontrivial content of Theorem 2.36(a) is to say that the global generating section is actually already a section of the presheaf $F_0$.
(3) Lemma 2.37 doesn't quite say that $F_0$ is a sheaf if $M$ is closed, because $A$ is required to be a compact set, rather than an open set. Here is an instructive example. Take $M=S^1$ and let $U\subset M$ be an open set whose complement is countably infinite. Then $U$ is a disjoint union of countably infinitely many open intervals, so $F(U)\cong R^\mathbb{N}$ (since $F(V)=R$ for any open interval $V\subset S^1$). But $F_0(U)$ can be computed directly to be a direct sum of countably infinitely many copies of $R$ (the key point being that $H_0(M-U)$ is the free $R$-module on $M-U$, which is countable). So $F_0(U)\not\cong F(U)$, so $F_0$ is not a sheaf.