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Model categories and topological quantum field theories. (English) Zbl 0864.57032

In the paper [Topological quantum field theories derived from the Kauffman bracket, Topology 34, No. 4, 883-927 (1995)] the first author and C. Blanchet, N. Habegger and P. Vogel construct the topological quantum field theory (TQFT) whose invariant is the Jones-Witten-Reshetikhin-Turaev invariant of 3-manifolds by means of the cobordism category \({\mathcal C}\) of 2-dimensional surfaces and 3-dimensional bordisms with structure. The category \({\mathcal C}\) has as objects all closed surfaces with structure. Thus it is not a small category: the objects of \({\mathcal C}\) do not form a set. On the other hand, it is known that all manifolds can be embedded into \(\mathbb{R}^\infty\), and that submanifolds of \(\mathbb{R}^\infty\) form a set. The aim of the paper under review is to describe how the cobordism category \({\mathcal C}\) considered in the paper cited above can be replaced by a small model category \({\mathcal M}\) of manifolds and cobordisms embedded in \(\mathbb{R}^\infty\), without losing any information about the TQFT-functors.
Reviewer: V.Abramov (Tartu)

MSC:

57R99 Differential topology
53C80 Applications of global differential geometry to the sciences
81T99 Quantum field theory; related classical field theories
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