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- Holonomy and parallel transport for Abelian gerbesPublication . Mackaay, Marco; Picken, R.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 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 less than or equal to2. 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. (C) 2002 Elsevier Science (USA).
- Simple transitive 2-representations of small quotients of Soergel bimodulesPublication . Kildetoft, Tobias; Mackaay, Marco; Mazorchuk, Volodymyr; Zimmermann, JakobIn all finite Coxeter types but I-2(12), I-2(18), and I-2(30), we classify simple transitive 2-representations for the quotient of the 2-category of Soergel bimodules over the coinvariant algebra which is associated with the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive 2-representations are exhausted by cell 2-representations. However, in Coxeter types I-2(2k), where k >= 3, there exist simple transitive 2-representations which are not equivalent to cell 2-representations.
- Spherical 2-categories and 4-manifold invariantsPublication . Mackaay, Marco
- Finite groups, spherical 2-categories, and 4-manifold invariantsPublication . Mackaay, MarcoIn this paper we dc line a class of state-sum invariants of closed oriented piece wise lineal 4-manifolds using finite groups. The definition of these state-sums Follows from the general abstract construction of 4-manifold invariants using spherical 2-categories. as we defined in an earlier paper. We show that the state survival invariants of Birmingham and Ratowski, who Studied Dijkgraaf Witten type invariants in dimension 3, are special examples of the general construction that we present in this paper. They showed that their invariants are non-trivial by some explicit computations, so our construction includes interesting examples already. Finally, wt indicate how our construction is related to homotopy 3-types. This connection suggests that there ale many more interesting examples of our construction to be found in the work on homotopy 3-types, by Brown, For example. (C) 2000 Academic press.
- Simple transitive 2-representations via (Co-)Algebra 1-MorphismsPublication . Mackaay, Marco; Mazorchuk, Volodymyr; Miemietz, Vanessa; Tubbenhauer, DanielFor any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using co-algebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also be constructed using algebra 1-morphisms in the projective abelianization of C. We also extend Morita-Takeuchi theory to our setup and work out several examples, including that of Soergel bimodules for dihedral groups, explicitly.