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13
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... 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 ..."
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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
Tensor categories attached to double groupoids
 Adv. Math
, 2006
"... Abstract. The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and q ..."
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Cited by 5 (2 self)
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Abstract. The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and quantum and FrobeniusPerron dimensions of simple objects are computed. Contents
EXACT SEQUENCES OF FIBRATIONS OF CROSSED COMPLEXES, HOMOTOPY CLASSIFICATION OF MAPS, AND NONABELIAN EXTENSIONS OF GROUPS
"... The classifying space of a crossed complex generalises the construction of EilenbergMac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy ..."
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The classifying space of a crossed complex generalises the construction of EilenbergMac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CWcomplex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory.
THE FUNDAMENTAL CROSSED MODULE OF THE COMPLEMENT Of A Knotted Surface
, 2009
"... We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, the ..."
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We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) →Gcan be rescaled to a homotopy invariant IG(M), depending only on the algebraic 2type of M. We describe an algorithm for calculating π2(M,M (1) ) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0 and 1handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a nontrivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. Th ..."
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Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.
On the Homotopy Type and the Fundamental Crossed Complex of the Skeletal Filtration of a CWComplex
, 2008
"... We prove that if M is a CWcomplex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with ndisks Dn, when the later are given their natural CWdecomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H ..."
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We prove that if M is a CWcomplex, then the homotopy type of the skeletal filtration of M does not depend on the cell decomposition of M up to wedge products with ndisks Dn, when the later are given their natural CWdecomposition with unique cells of order 0, (n − 1) and n; a result resembling J.H.C. Whitehead’s work on simple homotopy types. From the Colimit Theorem for the Fundamental Crossed Complex of a CWcomplex (due to R. Brown and P.J. Higgins), follows an algebraic analogue for the fundamental crossed complex Π(M) of the skeletal filtration of M, which thus depends only on the homotopy type of M (as a space) up to free product with crossed complexes of the type Dn. = Π(Dn),n ∈ N. This expands an old result (due to J.H.C. Whitehead) asserting that the homotopy type of Π(M) depends only on the homotopy type of M. We use
2003b, ‘Category Theory and Higher Dimensional Algebra: Potential Descriptive Tools in Neuroscience
 Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, National Brain Research Centre, Conference Proceedings
"... We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as potential yoga for dealing with processes and processes of processes. ..."
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We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as potential yoga for dealing with processes and processes of processes.
THREE THEMES IN THE WORK OF CHARLES EHRESMANN: LOCALTOGLOBAL; GROUPOIDS; HIGHER DIMENSIONS.
"... Abstract. This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. Introduction It is a pleasure to honour Charles Ehresmann by ..."
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Abstract. This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. Introduction It is a pleasure to honour Charles Ehresmann by giving a personal account of some of the major themes in his work which interact with mine. I hope it will be useful to suggest how these themes are related, how the pursuit of them gave a distinctive character to his aims and his work, and how they influenced my own work, through his writings and through other people.
COMPLICIAL SETS
, 2004
"... Abstract. The primary purpose of this work is to characterise strict ωcategories as simplicial sets with structure. We prove the StreetRoberts conjecture which states that they are exactly the “complicial sets ” defined and named by John Roberts in his handwritten notes of that title [26].2 VERITY ..."
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Abstract. The primary purpose of this work is to characterise strict ωcategories as simplicial sets with structure. We prove the StreetRoberts conjecture which states that they are exactly the “complicial sets ” defined and named by John Roberts in his handwritten notes of that title [26].2 VERITY
THE STRUCTURE OF DOUBLE GROUPOIDS
, 2008
"... Abstract. We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2cocycles with coefficients in abelian group bundles. Our description goes as follows: To any double groupoid, we ..."
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Abstract. We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2cocycles with coefficients in abelian group bundles. Our description goes as follows: To any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a ‘slim ’ double groupoid. In a first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In a second, independent, step we prove that every slim double groupoid with filling condition is completely determined by a factorization of a certain canonically defined ‘diagonal ’ groupoid.