Here's a somewhat bizarre question, out of curiosity.
When dealing with concatenation of curves, it's convenient to define them in arbitrary intervals, so that the concatenation of curves defined in $[a,b]$ and $[b,c]$ is automatically a curve defined in $[a,c]$.
Nevertheless, when dealing with homotopies between curves, it's convenient to define them all in a standardized interval, for example $[0,1]$, so that homotopies can be easily defined in the square $[0,1]\times[0,1]$.
Question: is there a notion of 'trapezoidal Cartesian product', i.e., something analogous to the Cartesian product, but more suitable for dealing with homotopies between curves that are defined in distinct intervals?
(Just for clarity: of course, there's an obvious way to solve the problem: just bring all curves to $[0,1]$ before considering homotopies between them; that's what's usually done, as mentioned in the first paragraphs. Though, I'd like to know if there's a most symmetric possible way to do it, and if this symmetry can be relevant in other contexts rather than the simple ones we meet at initial algebraic topology courses.)
It seems to me that such a notion could possibly be obtained by simultaneously considering a homotopy between curves, and a homeomorphism (or perhaps a homotopy equivalence?) between the intervals where these curves are defined.
Besides that, I have a feeling that this might be easily done, if possible, by means of some categorical limit; and, moreover, that it might even be employed, if useful, in more complicated contexts of homotopical algebra. Well... if it's possible and useful, then being a rather simple idea it should already exist, but I didn't hear about it yet; at least not in a form I was able to recognize.
PS. I believe I could invent some definition, but since this kind of activity, when intended by people who are not yet experienced in the area, often results in totally sterile notions, I thought it would be more reasonable to ask before trying!
Edit: While reviewing this question, as suggested by MSE, I realized the answer might possibly be related to the graph of a homeomorphism between distinct intervals, via composition with the canonical projections. Though, I decided to publish it anyway because the possibility of using a homotopy equivalence instead of a homeomorphism seems to be more subtle, and because the question and any aswers might be interesting to someone else.
In abstract homotopy theory we have a much more general notion of homotopy:
Definition. Given morphisms $f_0, f_1 : X \to Y$, a left homotopy consists of the following data:
An object $\tilde{X}$ and weak equivalences $i_0, i_1 : X \to \tilde{X}$, for which there exists a morphism (necessarily a weak equivalence) $r : \tilde{X} \to X$ such that $r \circ i_0 = r \circ i_1 = \textrm{id}_X$.
A morphism $\tilde{f} : \tilde{X} \to Y$ such that $f \circ i_0 = f_0$ and $f \circ i_1 = f_1$.
The traditional notion of homotopy in topological spaces is the special case where we require $\tilde{X} = X \times [0, 1]$ and $i_0 (x) = (x, 0)$ and $i_1 (x) = (x, 1)$.
The general definition has its advantages and disadvantages. One advantage is that we don't have to privilege any kind of interval object. One disadvantage is that there is no obvious way to glue left homotopies together. (In a model category, this is possible for a slightly less general notion of left homotopy.) Also, the notion of weak equivalence is taken as given – so this is not suitable if you want to define weak equivalences to be homotopy equivalences and haven't yet defined what a homotopy equivalence is.