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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
Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
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Cited by 3 (2 self)
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Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
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
RESEARCH STATEMENT
"... My reseach interests lie in algebraic topology and homotopy theory, and my doctoral work is focused on the realization of Πalgebras. More precisely, my goal is to apply the obstruction theory of BlancDwyerGoerss [9] to the case of truncated Πalgebras. Since the obstructions to existence and uniq ..."
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My reseach interests lie in algebraic topology and homotopy theory, and my doctoral work is focused on the realization of Πalgebras. More precisely, my goal is to apply the obstruction theory of BlancDwyerGoerss [9] to the case of truncated Πalgebras. Since the obstructions to existence and uniqueness live in certain Quillen cohomology groups of Πalgebras, my main focus has been the algebraic problem of understanding these groups better and trying to compute them. 1. Background 1.1. Πalgebras and the realization problem. One of the main ideas of algebraic topology is to describe spaces by associating algebraic invariants to them. The homotopy groups πi(X) of a space X are an important example and have been studied extensively. They are a collection of groups (abelian if i> 1), but they also carry additional structure: a π1action on higher groups, Whitehead products πi × π j → πi+ j−1, and primary homotopy operations corresponding to precomposition by maps between spheres, i.e. πn(S j) × π j → πn. This algebraic structure is known as a Πalgebra and was formally introduced in [20]. The prototypical Πalgebra is the homotopy Πalgebra π∗(X) of a
ON THE NON–BALANCED PROPERTY OF THE CATEGORY OF CROSSED MODULES IN GROUPS.
, 2003
"... Abstract. An algebraic category C is called balanced if the cotriple cohomology of any object of C vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of nonbalanced categories occur in the literature. In this paper we prove that the category of cros ..."
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Abstract. An algebraic category C is called balanced if the cotriple cohomology of any object of C vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of nonbalanced categories occur in the literature. In this paper we prove that the category of crossed modules in groups is nonbalanced.