We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) → C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

We generalize the notion of identities among relations, well known for presentations of groups, to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that th...

Given a categorical crossed module H→ G, where G is a group, we show that the category of derivations, Der(G,H), from G into H has a natural monoidal structure. We introduce the Whitehead categorical group of derivations as the Picard category of Der(G,H) and then we characterize the invertible derivations, with respect to the tensor product, in this monoidal category. 2000 Mathematics Subject ...

The authors have used generalised Galois Theory to construct a homotopy double groupoid of a surjective fibration of Kan simplicial sets. Here we apply this to construct a new homotopy double groupoid of a map of spaces, which includes constructions by others of a 2-groupoid, cat-group or crossed module. An advantage of our construction is that the double groupoid can give an algebraic model of...

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)-connected (n + 1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secondary ...

Abstract We study higher-form global symmetries and a higher-group structure of low-energy limit (3 + 1)-dimensional axion electrodynamics in gapped phase described by topological action. argue that the should have semi-strict 4-group (3-crossed module) consistency conditions couplings action to background gauge fields for symmetries. find possible ’t Hooft anomalies symmetry, discuss physical ...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.