I am reading this PhD thesis and I can't understand definition 6.21:
(Hodge groupoid). $(X/S)_{Hod}$ is a groupoid whose object object is $X\times \Bbb A^1_S$ and whose morphism object $N_{Hod}$ is given by the following procedure. First, blow up $X \times_S X \times_S \Bbb A^1 S$ at $∆ \times 0$. Next, take the complement of the strict transform of $X \times X \times_S 0$. Finally, take the PD completion along $∆ \times_S \Bbb A^1$.
I don't understand what is the "object object" and "morphism object". Does it mean the objects and the morphisms of the fiber over and object in the base category (since we deal with stacks)?. I have some notions of category theory and for me there are "objects" and "isomorphisms" in a groupoid. I am mainly interested in understanding what $N_{Hod}$ represents in the Moduli stack of integrable $\lambda$-connections (denoted $M_{Hod}$) in this thesis. In particular, I want to know if Proposition $6.22$ ($N_{Hod}$ is flat over $\Bbb A^1$) means that $M_{Hod}$ is flat over $\Bbb A^1_S$?
Thank you for your lights.