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 $\mathcal{X}$?
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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 $\mathcal{X}$? |
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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 $\mathcal{X}$? |
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