Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 315ders. A surgery gives new (co)bordism: from M(A, B) into N(A, B). Thedisjoint sum of M(A, B) with N(C, D) is a (co)bordism (M ∪ N)(A ∪C, B ∪ D). We got a 2–graph of (co)bordism Cob with Cob 0 = Man d ,Cob 1 = Man d+1 , whose 2–cells from Cob 2 are surgery operations.There is an n−category of (co)bordisms BO [Leinster (2003)] with:• 0−cells: 0−manifolds, where ‘manifold’ means ‘compact, smooth, orientedmanifold’. A typical 0−cell is • • • • .• 1−cells: 1−manifolds with corners, i.e., (co)bordisms between0−manifolds, such as(this being a 1−cell from the4−point manifold to the 2−point 0−manifold).• 2−cells: 2−manifolds with corners, such as• 3−cells, 4−cells,... are defined similarly;• Composition is gluing of manifolds.The (co)bordisms theme was taken a step further by [Baez and Dolan(1995)], when when they started a programme to understand the subtlerelations between certain TMFT models for manifolds of different dimensions,frequently referred to as the dimensional ladder. This programme isbased on higher–dimensional algebra, a generalization of the theory of categoriesand functors to n−categories and n−functors. In this framework atopological quantum field theory (TMFT) becomes an n−functor from then−category BO of n−cobordisms to the n−category of n−Hilbert spaces.