Question concerning the notation of path multiplication I'm self-studying topology by using Lee's Introduction to topological manifolds. I've just started reading the chapter on Homotopy and the Fundamental Group. Untill now everything makes perfect sense to me. The only thing bothering me is the notation for the path multiplication, which is
\begin{equation}
fg(s):=\begin{cases}f(2s), s\in [0,1/2] \\
g(2s-1), s \in [1/2,1]\end{cases}
\end{equation} first going along $f$ and then $g$. Because composition of functions $f:X\to Y, g:Y \to Z$ has always been denoted by $g\circ f:X\to Z$ I'd find it much more compatible to denote the path multiplication by $gf$ instead of $fg.$ 
So my question is why this notation is used?

One reason for the notation is the prevalence of graphical techniques used for gaining intuition, teaching or even as mathematical proof. For instance the following diagram represents the homotopy giving associativity for loop multiplication $(f\cdot g)\cdot h\simeq f\cdot (g\cdot h)$

In the diagram the left-most region represents the loop $f$, the middle region represents the loop $g$ and the right-hand region represents $h$. Clearly it displays from left to right and leads to adopting the notation $f\cdot g \cdot h$ for the loop $f$ followed by $g$ followed by $h$.
And if you're not happy with that answer then do what I would normally do and blame it all on poor old Henri Poincaré, http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf page 58.
