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added these pointers:
A more geometric model of orbifold K-theory in terms of bundles of Fredholm operators over Lie groupoids/differentiable stacks:
Review in:
{#Freed12} Daniel Freed, Lecture 1 of: Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (FreedESI2012.pdf:file)
Joost Nuiten, Section 3.2.2 of: Cohomological quantization of local prequantum boundary field theory MSc thesis, Utrecht, August 2013 (pdf)
The claim that these two definitions are equivalent, in that this groupoid K-theory reduces to equivariant K-theory on global quotient orbifolds, is Freed-Hopkins-Teleman 07, Prop. 3.5.
I don’t know where you want to put this new article, but it looks relevant:
Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory, https://arxiv.org/abs/2006.12374
It’s from a student of Schwede. I haven’t seen you mention it, so apologies if you have seen this already.
Thanks, I had missed that.
Okay, I have added the pointer to the entry, as follows:
The suggestion (Schwede 17, Intro, Schwede 18, p. ix-x) that orbifolds should be regarded as orbispaces in global equivariant homotopy theory and then their orbifold cohomology be given by equivariant cohomology with coefficients in global equivariant spectra is worked out for (Bredon cohomology and) orbifold K-theory in:
Example 5.31 there shows that on global quotient orbifolds this is again equivalent to the previous definitions.
I have added pointer to the “full orbifold K-theory” of
They prove it agrees with Adem-Ruan (and hence with all other definitions) on global quotients. I guess this means it agrees with Freed-Hopkins-Teleman and Juran in general? Does anyone discuss this?
Added pointer to
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