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$\DeclareMathOperator\PGL{PGL}$The classifying stack of $\PGL(2)$ is the stack quotient $[\operatorname{Spec} k/\PGL(2)]$ where $\PGL(2)$ acts trivially on $\operatorname{Spec} k$.

Since $[\operatorname{Spec} k/\PGL(2)]$ is a quotient stack, it is an algebraic stack with smooth covering map $q: \operatorname{Spec} k \to [\operatorname{Spec} k/\PGL(2)]$ defined by the pair $(\PGL(2), \rho)$ where $\rho: \PGL(2) \times \operatorname{Spec} k \to \operatorname{Spec} k $ is the action map.

On the other hand, $[\operatorname{Spec} k/\PGL(2)]$ is also the algebraic moduli stack of genus-zero curves. Since any map $\operatorname{Spec} k \to [\operatorname{Spec} k/\PGL(2)]$ is identified by Yoneda's lemma with an object in the fiber over $[\operatorname{Spec} k/\PGL(2)](\operatorname{Spec} k)$ a smooth covering map $q':\operatorname{Spec} k \to [\operatorname{Spec} k/\PGL(2)]$ could just as well have been defined by $\mathbb{P}^1$.

How can I recover the map $q$ defined by $(\PGL(2), \rho)$ from the smooth covering map $q'$ defined by $\mathbb{P}^1$?

I know that a map $\operatorname{Spec} k \to [\operatorname{Spec} k/\PGL(2)]$ should correspond to more than just an object in $[\operatorname{Spec} k/\PGL(2)](\operatorname{Spec} k)$—maybe an object plus automorphism of that object.

Improve this question
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    $\begingroup$ The genus 0 curve P^1 classifies a map $q’ : Spec(k) \to M_{0,0}$. This map is a trivial PGL(2)-torsor and exhibits $M_{0,0}$ as the quotient of $Spec(k)$ by the quotient by the trivial PGL(2)-action. In other words, the quotient map $q$ in terms of $q’$ is $Spec(k) \xrightarrow{q’} M_{0,0} \cong [Spec(k)/PGL(2)]$. $\endgroup$
    – crystalline
    Apr 18, 2021 at 0:49
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    $\begingroup$ Last para - if X is a stack then maps $Spec(k) \to X$ do correspond to objects of $X(Spec(k))$. “Object plus automorphism” is classified by the inertia stack of X $\endgroup$
    – crystalline
    Apr 18, 2021 at 0:51

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