'Composition' is the standard word used in the literature. There are two perspectives you might take on this: Firstly, the most mundane reason is that this is initially defined so that we can define the operation for fundamental groups. Secondly, (and secretly encompassing the first reason) we can define the fundamental groupoid of a space to be the category whose objects are points in the space and whose morphisms from a point x to a point y are endpoint-preserving homotopy classes of paths from x to y. In this case composition of morphisms is precisely the thing described above. I will hopefully do a video on this at some point, but in the meantime, if you are interested you can read more in chapter 2 of 'A Concise Course in Algebraic Topology' available here: www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
concatenation might be a more intuitive description?
What is the requirement on I? Should it be a Q-vector space? An order topology?
I is usually unit interval [0,1], at least conventioally in this context.
I can't understand, when u said that portion f double speed, g double speed.. can u plz elaborately explain that?
Composition is wrong word because range of g is different from domain of f, I think product of path is wright word
'Composition' is the standard word used in the literature. There are two perspectives you might take on this:
Firstly, the most mundane reason is that this is initially defined so that we can define the operation for fundamental groups.
Secondly, (and secretly encompassing the first reason) we can define the fundamental groupoid of a space to be the category whose objects are points in the space and whose morphisms from a point x to a point y are endpoint-preserving homotopy classes of paths from x to y. In this case composition of morphisms is precisely the thing described above. I will hopefully do a video on this at some point, but in the meantime, if you are interested you can read more in chapter 2 of 'A Concise Course in Algebraic Topology' available here: www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf