Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor -- categorical groups, groupoids, Lie algebroids and their higher analogues -- appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where infinity-groupoids appear in the role both of the domain and the coefficient object. Such cocycles in particular represent higher principal bundles, gerbes, -- possibly equivariant, possibly with connection -- as well as the corresponding associated higher vector bundles. We show how the Hopf algebra known as the Drinfeld double arises in this context. This article is an expansion of a talk that the second author gave at the 5th Summer School of Modern Mathematical Physics in 2008.
Deep Dive into Categorified symmetries.
Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of categorical flavor – categorical groups, groupoids, Lie algebroids and their higher analogues – appear in physically motivated constructions and faciliate constructions of geometrically sound models and quantization of field theories. Here we consider two flavours of categorified symmetries: one coming from noncommutative algebraic geometry where varieties themselves are replaced by suitable categories of sheaves; another in which the gauge groups are categorified to higher groupoids. Together with their gauge groups, also the fiber bundles themselves become categorified, and their gluing (or descent data) is given by nonabelian cocycles, generalizing group cohomology, where infinity-groupoids appear in the role both of the domain and the coefficient ob
The first part of this article is an overview for a general audience of mathematical physicists of (some appearances of) categorified symmetries of geometrical spaces and symmetries of constructions related to physical theories on spaces. Our main emphasis is on geometric and physical motivation, and the kind of mathematical structures involved. Sections 2-4 treat examples in noncommutative geometry, while 5-6 introduce nonabelian cocycles motivated in physics.
In sections 6-9 we discuss some technical details concerning differential cocycles and their quantization; part of these sections can be understood as a research anouncement.
Warning on versions: The original version of this article has been submitted in December 2008, and appeared in 5th Summer School of Modern Mathematical Physics, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dragovich, Zoran Rakić). In this arXiv version we have slightly updated some introductory points, and in particular the subsections 2.5. and 7.2. are entirely new. Section 7.1. on connections on principal ∞-bundles is also new and serves to provide some more background for the examples in section 9, where for instance the discussion of the electromagnetically charged quantum particle from a categorical perspective is new and the whole subsection 9.7. on Chern-Simons theory. We have also appended the list of additional (mainly new) references alphabetically just below the original references . One should especially mention the important reference arXiv:0905.0731 [FHLT] which touches on similar issues of categorical foundations of quantum physics as the the work sketched here and in the larger manuscript [36], also from December 2008. We should also note that since publication many aspects of this and related work were discussed or presented in the online project nlab [66] in which we are participating.
We assume the reader is familiar with basics of the theory of categories, functors and sheaves, as the mathematical physics community has adopted these by now. At a few places for instance we use (co)limits in categories. Readers familiar with enriched and higher category theory ([3, 24, 65, 72]) can skip this subsection.
The concept of a category C is often extended in several directions [2,4,24], leading to the internal categories, internal groupoids, monoidal categories, enriched categories, strict n-categories, and various flavours of weak higher categories. We will just sketch the terminology for orientation.
Instead of a set C 1 = ObC of objects and set C 0 = MorC of morphisms, with the usual operations (assignment of identity i : X → id X to X; domain (source) and codomain (target) maps s, t : C 1 → C 0 ; composition of composable pairs of morphism • : C 1 × C 0 C 1 → C 1 ) one defines an internal category in some ambient category A by specifying object of objects C 0 and object of morphism C 1 which are both objects in A, together with morphisms i, s, t, • as above, and satisfying analogous relations. An internal groupoid is an internal category equipped with an inverse-assigning morphism (•) -1 : C 1 → C 1 satisfying the usual properties. For instance smooth groupoids (Lie groupoids) are internal groupoids in the category of manifolds [2,14,23,40]). A category may be given additional structure, e.g. a monoidal category is equipped with tensor (monoidal) products and tensor unit object (cf. [4,24,29] and section 3.). Given a monoidal category D, a D-enriched category C has a set of objects, but each set of morphisms hom C (A, B) is replaced by an object D in D; it is required that the composition be a monoidal functor. In particular D may be the category of small categories, in which case a D-enriched category is precisely a 2-category: it has morphisms between morphisms. This process may be iterated and leads to n-categories of various flavour, with n-morphisms or n-cells as morphisms between (n -1)-morphisms. A strict (n + 1)-category is the same as nCat-enriched category where nCat is the category of strict n-categories and strict n-functors. If the cells for all n ≥ 0 are allowed we are dealing with ω-categories.
It is natural to weaken the associativity conditions for compositions of k-cells for 0 < k < n. This weakening is difficult to deal with, and there are multiple definitions, but this weakening is often naturally arising in applications and is more natural from the point of view of category theory itself. Thus one can talk about weak n-categories [3,24,65,72].
The weakening is much easier if the higher cells are invertible -these are by definition the (n, 1)-categories in the sense of Baez and Dolan, including the case of (∞, 1)-categories, which are of central importance in applications. More generally, we may talk on (n, k)-categories, of (in general, weak) n-categories only r-cells for r > k are invertible, and in particular of (∞, k)-categories.
According to Grothendieck’s homotopy hypothesis (from [58], ex-plained also
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