2-Categories, 4d state-sum models and gerbes

Differentiable Categories, gerbes and G-structures

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose …

Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, which acts on generic morphism categories of bundle gerbes. We continue by presenting a categorification of the hermitean metric on a hermitean line bundle. This is achieved by…

Lie范畴基础与乘法Ehresmann联络的Yang-Mills理论

Fundamentals of Lie categories and Yang-Mills theory for multiplicative Ehresmann connections

本文研究Lie范畴的基本性质，并将Yang-Mills理论从主丛推广到一般Lie群胚与代数胚，通过乘法Ehresmann联络、障碍类和变分原理，得到不可积非传递情形下的一对规范不变方程，并以$S^1$-丛gerbes为例加以说明。