I'm learning homology and cohomology by myself, and I've stumbled upon a nice introductory paper here (http://www3.nd.edu/~mbehren1/18.904/Heffern_project.pdf). On page 1, under Motivation, second paragraph, the authors say the following:
"We can have a function $f$ that defines the height of each vertex of the complex as an integer; this function extends naturally to a homomorphism $f \colon \Delta_{0}(X) \longrightarrow \mathbb{Z}$."
My questions are:
1) What is the height of a vertex? I know of the height of a tree, but I've never heard of such a thing for a single vertex.
2) How can this extend to a homomorphism? Particularly, what is the natural operation in $\Delta_{0}$?
Long story short, I'm somewhat comfortable with examples from simplicial homology. Now, to better familiarize myself with cohomology, I am trying to find concrete and detailed examples in which the homomorphisms from, say, $\text{Hom}(\Delta_{i}, \mathbb{Z})$, are explicit. Naturally I have a third question:
3) What other groups (aside from $\mathbb{Z}$) make sense to map $n$-simplices to?
As far as I understood from this paper the height function is just any function from the set of vertices to integers.
$\Delta_0$ is a free abelian group generated by vertices and for every free group $G$ with a set $S$ of generators any function $S\to H$, where $H$ is a group extends uniquely to a group homomorphism $G\to H$.
The elements of $\Delta_0$ are formal sums $\sum\limits_{x - vertex}xg_x$, where $g_x$ is its ,,height'' and only finitely many $g_x$ are nonzero. This is not necessarily if the complex is finite.
You can take coefficients in any other abelian group while constructing (co)homology not only $\mathbb Z$.