Results 1 
4 of
4
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
Abstract

Cited by 35 (10 self)
 Add to MetaCart
Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.