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  • $\begingroup$ Are you asking for something like the following: Let $\Omega$ be the set of maps $I: C([0,1],M)$ satisfying an appropriate list of universal properties. Then, for each $I \in \Omega$, there exists a $1$-form $\omega$ such that, for each continuous $C: [0,1] \rightarrow M$, $I(C)$ is the line integral of $\omega$ along $C$. $\endgroup$ Commented Jul 16, 2020 at 14:34
  • $\begingroup$ I don't think that I want "there exists a $1$-form $\omega$"; I believe that I want "for all $1$-forms $\omega$" instead. Alexander Betts' answer below seems extremely close to what I had in mind, but I'll leave the question open for several more days, to give it a chance to collect as many useful answers as possible. $\endgroup$ Commented Jul 16, 2020 at 15:24
  • $\begingroup$ My point is that I believe you can define the universal properties so that you do not even need to assume that what you are integrating $1$-forms along each curve $C$. The fact that the integrand is a $1$-form is already implied by the universal properties, if properly chosen. I think what I am proposing is even more general than what you are asking for. $\endgroup$ Commented Jul 16, 2020 at 19:28
  • $\begingroup$ Since I do not fully get your ideas, I have edited my question to hopefully make clearer what I am looking for. $\endgroup$ Commented Jul 17, 2020 at 8:00