Can we view an integral somehow as transfinite composition of “small” arrows, in a rigorous (but possibly nonstandard) way?

For example, if we view a real 1-form as a functor with values in $B\mathbb{R}$, is its integration along a curve a transfinite composition of its values at all the tangent vectors of the curve?

I don’t know if the question is clear enough, in case it isn’t, feel free to ask. (It may also be that none of this makes any sense.)

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