Formal definition of a $U(1)$ connection

Question

Let $\pi:P \rightarrow M$ be a $U(1)$ principal bundle. I often see people refer to a "$U(1)$ connection" but I cannot find a formal definition of this term. The closest I got was this question, but I'm still not entirely clear. The answer in that post says that a $U(1)$ connection is an object that locally looks like a 1-form and I am trying to understand what is meant by that. For example, if $V \subset M$ is a local trivialization of $P$, what is the definition of a $U(1)$ connection on $V$? Is it an element of $\Omega^1(V, X)$ for some space $X$? If so, what is $X$?

Answer 1

A $\mathrm{U}(1)$-connection on $P$ may be seen as a one-form $\omega\colon TP \to \mathfrak u(1) = \mathbb R$, where the latter is just the Lie algebra of $\mathrm{U}(1)$ (isomorphic to $\mathbb R$). The form $\omega$ must vanish on horizontal vector fields and must respect some equivariance property. But your question sounds more general than this, I think perhaps you should have a look at $G$-connections: https://en.wikipedia.org/wiki/Connection_(principal_bundle).