In Nakahara's *Geometry, Topology and Physics* on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an open covering $\{U_i\}$ on the base manifold $M$. Given a $\mathfrak{g}$-valued one-form $\mathcal{A}_i$ on $U_i$ and a local section $\sigma_i:U_i\to\pi^{-1}(U_i)$, he defines the connection one-form $\omega$ like this:

$$ \omega_i := g_i^{-1}\pi^*\mathcal A_i g_i + g_i^{-1}\mathrm{d}_Pg_i $$

Here, $\mathrm{d}_P$ denotes the exterior derivative on $P$ and $g_i$ is the canonical local trivialization such that $\phi_i^{-1}(u)=(p,g_i)$ and $u=\sigma_i(p)g_i$. Later he proves that $\sigma_i^*\omega_i=\mathcal A_i$, where he uses the fact that $\mathrm{d}_Pg_i({\sigma_i}_*X)=0$ because of $g\equiv e$ along ${\sigma_i}_*X$.

I don't understand the notation used here, especially the exterior derivative. As I understand it, the exterior derivative maps $p$-forms to $(p+1)$-forms, so for $\mathrm{d}_P$ to produce a one-form on $P$, it's input must be a 0-form, i.e. a function on $P$.

What's the function that serves as the "input" of $\mathrm{d}_P$? In other words, how does $g_i^{-1}\mathrm{d}_Pg_i$ produce a Lie-algebra-valued one-form?

connectionon a bundle, which is a well known and well established concept. $\endgroup$ – Alex Degtyarev Jun 25 '16 at 9:54