Smooth algebraic stacks with precisely two $\mathbb C$-objects In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\mathbb C$-object up to isomorphism. Unless I'm not mistaken, it is not hard to see that $\mathcal X$ is isomorphic to $BG$ for some affine finite type (smooth) group scheme $G$ over $\mathbb C$.
On the other hand, it seems natural to wonder about the following "exercise":
What is the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely two $\mathbb C$-objects up to isomorphism?
It could be the disjoint union of $BG$ with another $BG'$, but it could also be a connected stack such as $[\mathbb A^1/\mathbb G_m]$. Are these the only possibilities up to isomorphism? Or is there more?
 A: Here's what you can say (following my comments above).
Lemma. 
Let $X$ be a smooth finite type connected algebraic stack over $\mathbb C$ with precisely two $\mathbb C$-objects (up to isomorphism). Suppose that
1) the diagonal of $X$ is affine;
2) the stabilizers of all geometric points of $X$ are reductive; and
3) the stack $X$ has a dense open non-stacky point.
Then $X\cong [\mathbb A^1/\mathbb G_m]$.
Proof. This follows from the proof of the main theorem of Geraschenko-Satriano given in Toric Stacks II; see  arxiv.org/abs/1107.1907. Indeed, under our assumptions, $X$ has precisely one divisor. The natural open immersion associated to this Cartier divisor $X\to [\mathbb A^1/\mathbb G_m]$ is shown to be an isomorphism in their proof by using the local structure theorem of their paper (Theorem 4.5 in loc. cit.). QED
Remark. $[\mathbb P^1/\mathbb G_a]$ has precisely two objects, and it's not isomorphic to $[\mathbb A^1/\mathbb G_m]$ because condition 2) fails.
Remark. If you relax condition 3, then you can use "rigidification" to see that $X$ is a gerbe over $[\mathbb A^1/\mathbb G_m]$ for some reductive group $G$.
