# Proper domain and codomain of homotopies

I was studying topology when a question came to my mind.

Considering the definition of homotopies between (parallel) continuous functions, one notes that, in spite of their formal domain and codomain, conceptually they link continuous functions. I was wondering if there was a proper formal setting in which this could be highlighted, i.e. a category in which their domain and codomain would be the functions they link.

I was wondering about structuring C(X,Y) as a category with vertical composition of homotopies, but there I wouldn't know how to define identical homotopy for every continuous function, since the traditional identity homotopy of f (associating to every (x,t) the point f(x)) does not appear to be a neutral element for the composition above.

Update The situation looks similar to the one with paths in topological spaces: composing a path with a constant path in one of its extreme does not give the initial path, but only a path homotopic to it. The problem is solved in fundamental group considering classes of homotopic paths and a new composition among the latter classes. Should one apply this same viewpoint?

• Are you looking into homotopy theory, and in particular homotopy groups? Commented May 7 at 20:42
• @Arthur I'd like to keep it simple since I know only the basics of topology. I studied the fundamental group but among its relations with homotopy I know of I can't see one answering my question. Commented May 7 at 20:44
• What immediately comes to mind is the fundamental 2-groupoid of a space $X$: It's the 2-category whose objects are the points of $X$, whose 1-morphisms are paths in $X$ from one point to another, and whose 2-morphisms are path-homotopies, see the nlab for more information. Generally speaking this setting naturally gives rise to higher categorial descriptions: A homotopy is (or at least can be understood as) a "morphism between morphisms" (whatever this means), as your question notes. Commented May 7 at 20:55
• @AmandaWealth Yes, that would be the right path. I was not careful in my last comment wrt. this; one needs to replace "path-homotopies" by "homotopy classes of path-homotopies" for this to work. Asking what happens if one doesn't do this and instead further considers homotopies between homotopies, homotopies between homotopies between homotopies, etc. which leads to $\infty$-categories and the wonderful world of modern homotopy theory, but that seems somewhat out of scope for this question :) Commented May 7 at 21:10
• You are reinventing the homotopy 2-category of topological spaces. See this related question Commented May 7 at 21:12