space of connections is an affine space

Question

The space of connections on a principal $G$-bundle $E_G$ over the groupoid $\mathbb{X}=[X_1\rightrightarrows X_0]$ is an affine space for the space of all $\text{ad}(E_G)$-valued $1$-forms on the groupoid $\mathbb{X}=[X_1\rightrightarrows X_0]$.

Above statement is mentioned with out mentioning in what sense it is affine space.

Can some one spell out what does it mean to say? I am not asking for proof. Just the meaning of the above sentence..

In case you are not ok with Groupoids, read it as

The space of connections on a principal $G$-bundle $E_G$ over the manifold $M$ is an affine space for the space of all $\text{ad}(E_G)$-valued $1$-forms on the manifold $M$.

Can some one tell me what this statement mean?

Answer 1

I see that, by "The space of connections on a principal $G$-bundle $E_G$ over the manifold $M$ is an affine space for the space of all $\text{ad}(E_G)$-valued $1$-forms on the manifold $M$" it means the following:

Given two connection $1$-forms $\omega_1$ and $\omega_2$ on $E_G\rightarrow M$, the difference $\omega_1-\omega_2$ gives a $\text{ad}(P)$-valued $1$-form on $M$.

Answer 2

This means, if you fix a connection $a$. You can write any other connection $b=a+c$ where $c$ is the pullback to $E_G$ of an $ad(E_G) $ invariant 1 form by the projection map $E_G\rightarrow M$ and this correspondence is bijective.