Title
NonSemisimple Topological Quantum Field Theories for 3Manifolds with Corners (Lecture Notes in Mathematics, 1765),New
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d + 1dimensional manifold, whose is a union of ddimensional boundary disjoint v manifolds and d, a linear: + The manifold Zod V(Md+l) V(Zod) V(Zld). ma is with the orientation. The axiom in that z0g, Zod opposite gluing [Ati88] requires if we two such d + 1manifolds a common dsubma glue together along (closed) fold of in their the linear for the has to be the boundaries, composite compo map tion of the linear of the individual d + 1manifolds. maps the of and as in we can state categories functors, [Mac88], Using language axioms as follows: concisely Atiyah's very Definition 0.1.1 A in dimension d is a ([Ati88]). topological quantumfield theory between monoidal functor symmetric categories [Mac881 asfollows: V: ] kvect. Cobd+1 finite Here kvect denotes the whose are dimensional v category, objects for field tor over a field k, which we assume to be instance, a perfect, spaces The of of characteristic 0. set between two vector is morphisms, simply spaces the set of linear with the usual The has as composition. category Cobd+1 maps manifolds. such closed oriented ddimensional A between two objects morphism. Zd d oriented d 1 dmanifolds and is a + 1cobordism, an + Zod meaning gMd+l = Zd is the d mensional manifold, Md+l, whose Lj boundary _ZOd of the dmanifolds. consider union two we as joint (Strictly speaking morphisms cobordisms modulo relative Given another or homeomorphisms diffeomorphisms).
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