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(This question is duplicated here)

This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a (nonrepresentable) morphism to a point: $st:\mathcal{B}G\to P$. What do people mean when they say the "fundamental class" $[\mathcal{B}G]$ of $\mathcal{B}G$? (I don't know how homology carries over to stacks.) For example, is it true that $st_*[\mathcal{B}G]=[P]$? Or that $st^*[P]=[\mathcal{B}G]$? Are these equivalent?

My guess: It seems to me that the map $st$ should be considered as "degree" $1/|G|$. Because of this it seems like we should have $st^*[P]=|G|[\mathcal{B}G]$. I'm not sure what this would say about $st_*[P]$.

Thanks!

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    $\begingroup$ Rob- Instead of reposting, ask the moderators at math.SE to migrate your question. $\endgroup$
    – Ben Webster
    Feb 23, 2014 at 18:50
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    $\begingroup$ I recommend you consult the basic literature, e.g., MR1005008 (90k:14004), Vistoli, Angelo(1-HRV), Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), no. 3, 613–670. 14C17 (14A20 14D20) $\endgroup$ Feb 23, 2014 at 20:35
  • $\begingroup$ @BenWebster: Will do in the future, thanks! $\endgroup$ Feb 24, 2014 at 19:06

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