Timeline for Basic questions about stacks 2
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9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jun 23, 2011 at 23:16 | comment | added | David Roberts♦ | ...of stacks in groupoids on (S,J) is equivalent to the bicategory of groupoids, anafunctors and transformations in S. Roughly speaking, all fibres of a stack are representable in this bicategory, and so given WISC, one knows that the fibres are essentially small. Oh, I should mention that really this is only expected to hold for geometric stacks (presentable by some space/scheme/what-have-you), but I guess that that is the case you are most interested in. | |
| Jun 23, 2011 at 23:11 | comment | added | David Roberts♦ | Me again. The category GBund(X) of principal G-bundles on a fixed object X in a site (S,J) is equivalent to the hom-category Gpd_ana(X,_B_G) in the bicategory of internal groupoids, anafunctors and transformations (without loss of generality, assume S is a superextensive site - see nLab for definition - which is true in all geometric situations). Here _B_G is the groupoid with one object and morphisms G, and we consider X as a groupoid with only trivial morphisms. GBund(X) is essentially small when the axiom 'WISC' holds for J, again see nLab. But 'morally', if not actually, the 2-category.... | |
| Jun 23, 2011 at 14:14 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Jun 23, 2011 at 13:33 | answer | added | AAK | timeline score: 9 | |
| Jun 23, 2011 at 12:42 | comment | added | David Roberts♦ | Regarding 1), the definition of a fibred category does not assume that the fibres are small, but in geometric examples they are (or at least are essentially small). For example, would you consider the groupoid of G-bundles on a fixed topological space to be a set? This groupoid is certainly essentially small once you have a classifying space BG. | |
| Jun 23, 2011 at 10:04 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Jun 23, 2011 at 9:59 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Jun 23, 2011 at 9:46 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |