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There is a notion of $K$-theory for a manifold $M$.

Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $B\mathcal{G}\cong \mathcal{D}$?

One reference I could find for $K$-theory for Algebraic stacks is Bertrand Toen‘s thesis. Unfortunately, only title and abstract are in English and everything else is in French.

So, are there any references in English that introduce and discuss $K$-theory for stacks. What prerequisites would be needed to understand such theory for stacks?

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This is written nicely here. The the paper is called Loop Groups and Twisted $K$-theory I, by Freed, Hopkins and Teleman. Have a look at Section 3 and the Appendix.

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  • $\begingroup$ Thank you :) I do not have enough background for understanding the introduction... I will give one more try after reading somethings mentioned in introduction.. :) Thanks again for your answer $\endgroup$ Nov 27, 2019 at 5:46
  • $\begingroup$ You're welcome! There's no need to understand the stuff about loop groups, twistings etc. I guess you do need to be somewhat familiar with the Fredholm model for ordinary K theory $\endgroup$ Dec 2, 2019 at 4:39
  • $\begingroup$ Thanks for your reply... I know one or two things about K theory.. I will read about Fredholm model first :) Thanks :) $\endgroup$ Dec 2, 2019 at 6:44

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