I am attempting to start working through J.P. Mays A Concise Course in Algebraic Topology but can't seem to understand what it describes as "schematic indications" of how a given homotopy behaves on "the domain squares."
He first defines three path-functions: $$ F : x \to y, \qquad G : y \to z, \qquad H : z \to w. $$ and then notes that their compositions are associative -- I am sure there's a better way to put, he simply writes: $h \circ (g \circ f) \sim (h \circ g) \circ f$ -- this is also easy to believe since we "defined" composition of paths a bit earlier (by concatenating them and going "twice as fast", which I am more or less willing to accept for the moment).
He then defines a constant-loop path function $c(x) : x \to x$ and draws his picture of the "domain square". The top and bottom indicate the three paths we have defined, $f$, $g$, and $h$ -- and the left and the right indicate two of the identity functions, $c(x)$ and $c(w)$. He draws two slightly inclined line segments through the square, creative three volumes, which could now presumably be thought of as $f$, $g$ and $h$. Note that the bottom row doesn't say $f$-prime, it just says $f$; so I guess this I guess is where I am confused.
So I sort of get what this is saying -- these functions move points to other points in an 'orderly' way -- but I guess I am having trouble understanding what is intended by "domain square" and he does not really define it. The setup for some of this did reference metric spaces and some stuff about fundamental groups -- should that provide enough context to gather what the domain square is in this context and how to apply to decode these little schematic drawings? I would appreciate any help or clarification on these.
(In passing, and this is of course not necessary, if someone could suggest a slightly less, well, "concise" algebraic topology course book I would be grateful.)