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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 16th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting some minimum <a href="https://ncatlab.org/nlab/revision/orbifold+K-theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/orbifold+K-theory">current</a>

   starting some minimum

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 24th 2020
   • (edited Jun 24th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers: A more geometric model of orbifold K-theory in terms of [[bundles]] of [[Fredholm operators]] over [[Lie groupoids]]/[[differentiable stacks]]: * [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Loop groups and twisted K-theory I_, Journal of Topology, Volume4, Issue 4 ([arXiv:0711.1906](https://arxiv.org/abs/0711.1906), [doi:10.1112/jtopol/jtr019](https://doi.org/10.1112/jtopol/jtr019)) Review in: * {#Freed12} [[Daniel Freed]], Lecture 1 of: _Lectures on twisted K-theory and orientifolds_, lecures at _[K-Theory and Quantum Fields](http://www.esi.ac.at/activities/events/2012/k-theory-and-quantum-fields)_, ESI 2012 ([[FreedESI2012.pdf:file]]) * [[Joost Nuiten]], Section 3.2.2 of: _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_ MSc thesis, Utrecht, August 2013 ([pdf](http://ncatlab.org/schreiber/files/thesisNuiten.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](#FreedHopkinsTeleman07). <a href="https://ncatlab.org/nlab/revision/diff/orbifold+K-theory/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbifold+K-theory/2">v2</a>, <a href="https://ncatlab.org/nlab/show/orbifold+K-theory">current</a>

   added these pointers:

   A more geometric model of orbifold K-theory in terms of bundles of Fredholm operators over Lie groupoids/differentiable stacks:

   • Daniel Freed, Michael Hopkins, Constantin Teleman, Loop groups and twisted K-theory I, Journal of Topology, Volume4, Issue 4 (arXiv:0711.1906, doi:10.1112/jtopol/jtr019)

   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.

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 25th 2020
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, I had missed that.

   Thanks, I had missed that.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, I have added the pointer to the entry, as follows: The suggestion ([Schwede 17, Intro](orbifold+cohomology#Schwede17), [Schwede 18, p. ix-x](orbifold+cohomology#Schwede18)) 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: * Branko Juran, _Orbifolds, Orbispaces and Global Homotopy Theory_ ([arXiv:2006.12374](https://arxiv.org/abs/2006.12374)) Example 5.31 there shows that on [[global quotient orbifolds]] this is again equivalent to the previous definitions. <a href="https://ncatlab.org/nlab/revision/diff/orbifold+K-theory/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbifold+K-theory/3">v3</a>, <a href="https://ncatlab.org/nlab/show/orbifold+K-theory">current</a>

   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:

   • Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory (arXiv:2006.12374)

   Example 5.31 there shows that on global quotient orbifolds this is again equivalent to the previous definitions.

   diff, v3, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added pointer to the "full orbifold K-theory" of * Tyler J. Jarvis, Ralph Kaufmann, Takashi Kimura, _Stringy K-theory and the Chern character_, Invent. math. 168, 23–81 (2007) ([arXiv:math/0502280](https://arxiv.org/abs/math/0502280), [doi:10.1007/s00222-006-0026-x](https://doi.org/10.1007/s00222-006-0026-x)) 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? <a href="https://ncatlab.org/nlab/revision/diff/orbifold+K-theory/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbifold+K-theory/4">v4</a>, <a href="https://ncatlab.org/nlab/show/orbifold+K-theory">current</a>

   I have added pointer to the “full orbifold K-theory” of

   • Tyler J. Jarvis, Ralph Kaufmann, Takashi Kimura, Stringy K-theory and the Chern character, Invent. math. 168, 23–81 (2007) (arXiv:math/0502280, doi:10.1007/s00222-006-0026-x)

   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?

   diff, v4, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 30th 2021
   • (edited Jul 30th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded pointer to * [[Ernesto Lupercio]], [[Bernardo Uribe]], *Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. ([arXiv:math/0105039](http://arxiv.org/abs/math/0105039), [doi:10.1007/s00220-003-1035-x](https://doi.org/10.1007/s00220-003-1035-x)) <a href="https://ncatlab.org/nlab/revision/diff/orbifold+K-theory/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/orbifold+K-theory/11">v11</a>, <a href="https://ncatlab.org/nlab/show/orbifold+K-theory">current</a>

   Added pointer to

   • Ernesto Lupercio, Bernardo Uribe, Gerbes over Orbifolds and Twisted K-theory, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

   diff, v11, current