Spherical 2-Categories and 4-Manifold Invariants

Mackaay, Marco

doi:10.1006/aima.1998.1798

Cited by 55 publications

(115 citation statements)

References 30 publications

Supporting

Mentioning

114

Contrasting

Order By: Relevance

“…Let us briefly explain where they differ. The main motivation for our approach is that we are trying to understand the differential geometry behind the four-dimensional state-sum models defined by the first author of the present paper [22,21]. For the understanding of these state-sums it would be helpful to find a categorical way of perceiving the relation between homotopy theory and differential geometry.…”

Section: Introductionmentioning

confidence: 99%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

View full text Add to dashboard Cite

In this paper we establish a one-to-one correspondence between U (1)-gerbes with connections, on the one hand, and their holonomies, for simply-connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with * presently working as a postdoc at the University of Nottingham, UK 1 group U (1) on a simply-connected manifold M is a group morphism from the thin second homotopy group to U (1), satisfying a smoothness condition, where a homotopy between maps from [0, 1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply-connected case.

show abstract

Section: Introductionmentioning

confidence: 99%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…For weak versions of 2-groups, see [15]. A general framework for state sum invariants of four-manifolds is provided by Mackaay's construction [34].…”

Section: Technical Questionsmentioning

confidence: 99%

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

Pfeiffer

2003

Annals of Physics

102

View full text Add to dashboard Cite

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G = U (1), there exists a generalization, known as p-form electrodynamics, in which (p − 1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p = 2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.

show abstract

“…In their "n-categories with duals", they generalize the fact that the notion of adjoint functor can be expressed in 2-categorical terms within the 2-category 1CAT . Mackaay describes the application of internal adjoints to 4-manifold invariants in [157]. The notion of adjoint generalizes within an n-category to the notion of dual of any i-morphism for 0 < i < n. At the top level of n-morphisms, the dual operation should either be: ignored; imposed as additional structure; or pushed to ∞ by considering directly the theory of ∞-categories.…”

mentioning

confidence: 99%

Homotopy Theory of Higher Categories

Simpson

2011

111

View full text Add to dashboard Cite

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of n-nerve and Pelissier's thesis. If M is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of M -precategories. The procedure can then be iterated, leading to model categories of (∞, n)-categories.

show abstract

Spherical 2-Categories and 4-Manifold Invariants

Cited by 55 publications

References 30 publications

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

Homotopy Theory of Higher Categories

Contact Info

Product

Resources

About