Zoran Skoda Categorified Symmetries

Urs and just revised slightly our Proceedings paper published last year (and written in December 2008) to post it to the arXiv. Comments are very welcome! (We may try another iteration of changes later this month.) With the expanded material, less caring about bounds of the journal, with added table of contents and many more references it grew from 28 pages of the published version to 55 page of the current version.

arxiv/1004.2472

math.QA with crosslists hep-th, math.CT, math.AT

AMS classification:

• 81T Quantum field theory, related classical theories

• 14A22 Noncommutative algebraic geometry

• 18 Category theory

• 55N30 Sheaf cohomology (the closest approximation, shame on MSC2010 classification: stacks are mentioned only in subarea of algebraic geometry, not in general, word descent is nowhere, on various cohomologies of spaces and so on lots of MSC numbers, but none on general cohomology theory, no nonabelian cohomology, just nonabelian homological algebra what is not quite the same, no higher category MSC…name it)

The previously published version is

• Urs Schreiber, Zoran ?koda, Categorified symmetries, 5th Summer School of Modern Mathematical Physics, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dragovich, Zoran Raki?)

---

Its table of contents is:

Contents
1. Introduction 3
1.1. Categories and generalizations . . . . . . . . . . . . . . . . . 4
1.2. Basic idea of descent . . . . . . . . . . . . . . . . . . . . . . 5
2. From noncommutative spaces to categories 5
2.1. Idea of a space and of a noncommutative space . . . . . . . 5
2.2. Gel?fand-Naimark . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3. Nonaffine schemes and gluing of quasicoherent sheaves . . . 6
2.4. Noncommutative generalizations of QcohX . . . . . . . . . . 7
2.5. Abelian versus ∞-categories . . . . . . . . . . . . . . . . . . 7
3. Monoidal categories as symmetries of NC spaces 8
3.1. Basic appearances of Hopf algebras . . . . . . . . . . . . . . 8
3.2. A problem with tensor product . . . . . . . . . . . . . . . . 9
3.3. Replacing Hopf (co)actions with geometrically admissible actions of monoidal categories . . . . . . . . . . . . . . . . . . 9
3.4. Principal bundles on noncommutative schemes . . . . . . . 11
4. Application to Hopf algebraic coherent states 12
5. Higher gauge theories 13
6. ∞-Categories and homotopy theory 13
6.1. ∞-Categories versus model categories . . . . . . . . . . . . 14
6.2. Generalized spaces, topoi and (higher) categories . . . . . . 15
6.3. Strict ∞-Groupoid-valued ∞-stacks . . . . . . . . . . . . . . 16
7. Nonabelian cohomology, higher vector bundles and back- ground fields 17
7.1. Principal ∞-bundles . . . . . . . . . . . . . . . . . . . . . . 20
7.2. Associated ∞-bundles . . . . . . . . . . . . . . . . . . . . . 24
7.3. Sections of associated ∞-bundles . . . . . . . . . . . . . . . 26
7.4. Connections on ∞-bundles . . . . . . . . . . . . . . . . . . . 27
7.4.1. The homotopy ∞-groupoid . . . . . . . . . . . . . . 27
7.4.2. The geometric path ∞-groupoid . . . . . . . . . . . 31
7.4.3. Differential cocycles and connections . . . . . . . . . 33
8. Quantization and quantum symmetries 36
8.1. Background field and space of states . . . . . . . . . . . . . 37
8.2. Transgression of cocycles to mapping spaces . . . . . . . . . 37
8.3. Branes and bibranes . . . . . . . . . . . . . . . . . . . . . . 38
8.4. Quantum propagation . . . . . . . . . . . . . . . . . . . . . 39
9. Examples and applications 40
9.1. Ordinary vector bundles . . . . . . . . . . . . . . . . . . . . 40
9.2. The charged quantum particle . . . . . . . . . . . . . . . . . 41
9.3. Group algebras and category algebras from bibrane monoids 42
9.4. Monoidal categories of graded vector spaces from bibrane monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9.5. Twisted vector bundles . . . . . . . . . . . . . . . . . . . . . 43
9.6. Dijkgraaf-Witten theory . . . . . . . . . . . . . . . . . . . . 44
9.6.1. The 3-cocycle . . . . . . . . . . . . . . . . . . . . . . 45
9.6.2. Transgression of DWtheory to loop space: the twisted Drinfeld double . . . . . . . . . . . . . . . . . . . . . 47
9.6.3. The Drinfeld double modular tensor category from DW bibranes . . . . . . . . . . . . . . . . . . . . . . 48
9.6.4. The fusion product . . . . . . . . . . . . . . . . . . . 49
9.7. Outlook: Chern-Simons theory . . . . . . . . . . . . . . . . 50
10.Conclusion 51