Homotopy field theory in dimension 3 and crossed group-categories

By Vladimir Turaev. 2000

241 citations recorded by the source corpus.

A 3-dimensional homotopy quantum field theory (HQFT) can be described as a TQFT for surfaces and 3-cobordisms endowed with homotopy classes of maps into a given space. For a group $\pi$, we introduce a notion of a modular crossed $\pi$-category and show that such a category gives rise to a 3-dimensional HQFT with target space $K(\pi,1)$. This includes numerical invariants of 3-dimensional $\pi$-manifolds and a 2-dimensional homotopy modular functor. We also introduce and discuss a parallel notion of a quasitriangular crossed Hopf $\pi$-coalgebra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6bb27f69e512a21156310936fe4020339f536f18

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Monoidal Grothendieck Construction (2018, unknown): “In [CDL06], the authors introduce a category T of Turaev R-modules, as well as a category Z of Zunino R-modules, which serve as symmetric monoidal categories where group-(co)algebras and Hopf group-(co)algebras, [Tur00], live as (co)monoids and Hopf monoids respectively.”
Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category (2005, methodology): “We are ready now to introduce the desired braided T-category (we use terminology as in [18]; for the subject of Turaev categories, see also the original paper of Turaev [ 16 ] and [2], [17]).”
Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras (2004, background): “Key words: Hopf 2-cocycle, Galois objects, biGalois objects.”
On braided fusion categories I (2009, background): “The latter notion was introduced by Turaev [ 47 ].”
Modular Categories and Orbifold Models (2001, result): “5.7. Remark. The decomposition (5.2) and Theorem 5.5 together are equivalent to the statement that Rep A is a “G-crossed braided category” as defined in [ Tu ],”
Membrane Topology (2006, unknown): “Keywords: Algebraic Topology, Surfaces, Strings.”