Our vector $\mathbf{A}$ doesn't have to live in the same space as the coordinates, we can do this dance with some $V^{M'} = e^{M'}_{M}(x) V^{M}(x)$ and get a
$$U(x+dx,x)^M_N = \delta^M_N - dx^{\mu} (A_{\mu})^M_N,$$
where $A_{\mu}$ is a connection with matrix indices $M,N$. One should study the transformation law of a connection in both cases, and note that the $M,N=1$ case should reproduce electromagnetism gauge transformations, and in general Yang-Mills, the only issue is that the $\mu$ vs $M,N$ now transform differently. In the $M,N=1$ case we now obviously have $U(x,y) = e^{-\int A_{\mu}…