Here is Rotman's definition of the skyscraper sheaf: Let $A$ be an abelian group, $X$ a topological space, and $x \in X$. Define a presheaf by $x_*A(U) = \begin{cases} A & \text{if } x \in U,\\ \{0\} & \text{otherwise.} \end{cases}$ If $U \subseteq V$, then the restriction map $\rho_U^V$ is either $1_A$ or $0$.
He then goes on to say the stalks of $x_*(A)$ are $\{0\}$ except at $(x_*A)_x$ which is $A$.
I'm going to try to show this but I am not understanding what the stalks look like. So far in the book we look at sections that are continuous maps and so $[\sigma]$ would be a germ centered at $x$ in the stalk $\mathscr F_x$ and $\tau \in [\sigma]$ occurs when there exists an open set $W$ such that $\tau \vert_W = \sigma \vert_W$ (i.e., they agree on an open neighborhood of $x$).
But now our sections are just elements of either the abelian group $A$ or the abelian group $0$. What does restriction mean on a group element? (i.e., what does it mean to say that $\tau \in [\sigma] \in (x_*(A))_y$ where $x \neq y$? Looking at $\tau, \sigma$ as maps, it means there exists an open neighborhood of $y$ such that $\tau \vert_W = \sigma \vert_W$. But we just know these are elements of an abelian group. What does restriction on these elements mean?
Any clarification would be greatly appreciated. I'm eventually going to prove his last statement about what stalks look like in this sheaf, but want a better understanding of what germs in the stalk look like.
Note: There is another question on here about skyscraper sheafs and proving the above statement, but it does not really help me understand what stalks/sections/germs look like (I think it's a different definition)
Subquestion: If $P$ a presheaf, $x \in X$, $U \ni x$ and $\sigma \in P(U)$, what is $[\sigma] \in P_x$ look like? (in terms of direct limit?) What does it mean for $\tau \in [\sigma]$? In general (i.e., where $\tau, \sigma$ are not necessarily maps that we can restrict to an open set?)
Best Answer
The stalks of $x_*A$ are $\begin{cases} (x_*A)_y =A & \text {if}\: y \in \overline {\{x\}},\\ (x_*A)_y =\{0\} & \text {if} \: y \notin \overline {\{x\}} \end{cases}$
This follows immediately from the definition of a stalk and from the fact that every neighbourhood of $y$ contains $x$ if $y \in \overline {\{x\}}$, whereas in the second case there exists a neighbourhood of $y$ not containing $x$, namely $X\setminus \overline {\{x\}}$.
Edit
As an amusing exercise inspired by Martin's comment, try to convince yourself that in the case where $\overline {\{x\}}=X$ (a common situation in algebraic geometry, and we then say that $x$ is the generic point of $X$) the sheaf $x_*A$ equals the constant sheaf with stalk $A$ and thus definitely does not look like a proud sky-scraper but rather like a depressing US housing project or British council flat or French HLM ...