## On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups

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Venue: | Theory Appl. Categ |

Citations: | 10 - 0 self |

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@ARTICLE{Martins_onyetter’s,

author = {João Faria Martins and Timothy Porter},

title = {On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups},

journal = {Theory Appl. Categ},

year = {},

pages = {118--150}

}

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### Abstract

We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined

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Citation Context ...ned a 3-dimensional TQFT, which included the ω = 1 case of the Dijkgraaf-Witten Invariant, though he did not incorporate group cohomology classes into his TQFT. This construction appeared upgraded in =-=[Y2]-=-, where D.N. Yetter extended his framework to handle categorical groups. Recall that categorical groups are equivalent to crossed modules. They are often easier to consider when handling the objects a... |

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Citation Context ...and π2(B(G), ∗) → π2(B(H), ∗). By Theorem 1.18, this can be stated in algebraic terms. For the description of the associated model structure on the category of crossed modules, we refer the reader to =-=[BG]-=-. In particular, it follows that the reduced crossed modules G and H are weak equivalent if and only if B(G) and B(H) have the same 2-type. Since these CW-complexes are 2-types themselves, there follo... |

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Citation Context ... topological space, then S(M) is the singular complex of M. Since D is a Kan simplicial set, it follows that D is a strong deformation retract of S(|D|), a fact usually known as Milnor’s Theorem, see =-=[FP]-=-, 4.5. The map j1 is the geometric realisation of the inclusion SIMP(C, D) → SIMP(C, S|D|), and is therefore a homotopy equivalence. There exists a one-to-one correspondence between simplicial maps C ... |

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Citation Context ...duced crossed module. There exists a one-to-one correspondence between G-colourings of M and maps Π(M) → G, where M has the natural CW-decomposition given by its triangulation. This result appears in =-=[P1]-=- (Proposition 2.1). Note that any map f : Π(M) → G yields a G-colouring c f of M, where c f (K) = f(h(K)). Here K is a non-degenerate simplex of M. To prove this we need to use the Homotopy Addition L... |

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Citation Context ...2], where an extension of Yetter’s Invariant to handle general models of n-types appears. The construction of M. Mackaay in [Mk2] is related, conjecturally, with the n = 3 case of the construction in =-=[P2]-=-, with a further twisting by cohomology classes of 3-types, in the Dijkgraaf-Witten way. In [FM1, FM2], D.N. Yetter’s ideas were applied to 2-dimensional knot theory, and yielded interesting invariant... |

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Citation Context ...xplicit Description of the Low Dimensional Coboundary Maps Let G = (G, E, ∂, ⊲) be a reduced crossed module. We consider the U(1)cohomology of G. This is a very particular case of the construction in =-=[Pa]-=-. There was considered the general case of cohomology with coefficients in any π1(B(G))-module. From the discussion above, the group C 3 (G) of 3-cochains of G is given by all maps ω: G 3 × E 4 → U(1)... |

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Citation Context ...ulation of M. In particular f ∗ (ω) is well defined in the simplicial category for any continuous map f : M → B(G). The Simplicial Approximation Theorem (for simplicial sets) is proved for example in =-=[S]-=-. Note also the Normalisation Theorem stated in 2.1.1. The proof of Theorem 2.5 is analogous to the proof of Theorem 1.21. The main lemma which we will use for its proof is the following. 38Lemma 2.6... |

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Citation Context ...d action ⊲ of G on E by automorphisms. The conditions on ⊲ and ∂ are: 1. ∂(X ⊲e) = X∂(e)X −1 , if X ∈ G and e ∈ E are such that t(X) = β(e), 2. ∂(e) ⊲ f = efe −1 , if e, f ∈ E verify β(e) = β(f). See =-=[BI]-=-, section 1. Notice that for any e ∈ E we must have s(∂(e)) = t(∂(e)) = β(e). A crossed module is called reduced if P is a singleton. This implies that both G and E are groups. A morphism F = (φ, ψ) b... |

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Citation Context ...llary 1.20. In a future work, we will investigate the behaviour of IG under arbitrary crossed module maps G → H, in order to generalise this result. Yetter’s Invariant was extended to CW-complexes in =-=[FM2]-=-, where an alternative proof of its existence is given, using a totally different argument 24to the one just shown. Also in the cellular category, an extension of Yetter’s Invariant to crossed comple... |

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Citation Context ...ternative proof of its existence is given, using a totally different argument to the one just shown. Also in the cellular category, an extension of Yetter’s Invariant to crossed complexes appeared in =-=[FM3]-=-. In [FM1, FM2], algorithms 21are given for the calculation of Yetter’s Invariant of complements of knotted embedded surfaces in S 4 , producing non-trivial invariants of 2knots. Other similar interp... |

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Citation Context ...tement. Consider the complex C(D) = {Cn(D), ∂n} of simplicial chains of D. Here Cn(D) is the free abelian group on the set Dn of n-simplices of D. Furthermore ∂(c) = ∑n i=0 (−1)i∂i(c), if c ∈ Dn; see =-=[Ma]-=-, 1.2, or [We], 8.2. Note that the assignment D ↦→ C(D), where D is a simplicial set is functorial. The homology of a simplicial set D is defined as being the homology of the chain complex C(D). The c... |

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Citation Context ...specified, uniquely, by its value on the set of free generators of H. In other words, any map from the set of generators of H into E uniquely extends to a derivation d: H → E. Proof. This is shown in =-=[W1]-=- (Lemma 3) for the case in which H is a group, and is easy to prove. The general case can be proved directly in the same 20way. Alternatively, we can reduce it to the group case by using the free gro... |