6 deleted 2 characters in body

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibered fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ are the objects in $\mathcal{X}$ and morphisms of $\mathcal{X}^s$ are the morphisms in $\mathcal{X}$ modulo automorphisms of objects. It "kills" the groupoid structure, so I think it is possible to consider $\mathcal{X}^s$ as a category fibered fibred in sets over the category of schemes. Assume $\mathcal{X}^s$ is represented by a scheme. Should it be the coarse moduli space for $\mathcal{X}$?

5 deleted 17 characters in body

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibered in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ are (isomorphism classes of) the objects of in $\mathcal{X}$ and morphisms of $\mathcal{X}^s$ are the morphisms of in $\mathcal{X}$ modulo automorphisms of objects. It "kills" the groupoid structure, so I think it is possible to consider $\mathcal{X}^s$ as a category fibered in sets over the category of schemes. Assume $\mathcal{X}^s$ is represented by a scheme. Should it be the coarse moduli space for $\mathcal{X}$?

4 added 180 characters in body; added 1 characters in body

Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibered in groupoids over the category of schemes. Consider Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$. \mathcal{X}^s$are (isomorphism classes of) objects of$\mathcal{X}$and morphisms of$\mathcal{X}^s$are morphisms of$\mathcal{X}$modulo automorphisms of objects. It "kills" the groupoid structure, so I think it is possible to consider$\mathcal{X}^s$as a category fibered in sets over the category of schemes. Assume$\mathcal{X}^s$is represented by a scheme. Should it be the coarse moduli space for$\mathcal{X}\$?

3 deleted 212 characters in body
2 added 211 characters in body; edited title
1