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?



Exterior derivative on principal bundle
The expression $g^{-1} dg$ means the left invariant Maurer-Cartan 1-form on the Lie group $G$. It is only suggestive notation, but if $G$ is a subgroup of the general linear group $GL(n,\mathbb{R})$, then it has a precise meaning. The function $g : G \to \mathbb{R}^{n \times n}$ is the identity map, and then $dg : TG \to \mathbb{R}^{n \times n}$, and $g^{-1} dg$ is a 1-form on $G$ valued in $\mathfrak{g}$. So $g$ in his notation is the projection to $G$ from the local trivialization.


A clarification: in Nakahara, $g\_i^{-1} d\_P g\_i$ really denotes the pullback of the Maurer-Cartan form through the map $g\_i:\pi^{-1}(U\_i)\to G$


