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25
On algebraic models for homotopy 3types
 J. Homotopy Relat. Struct
"... We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2. ..."
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We explore the relations among quadratic modules, 2crossed modules, crossed squares and simplicial groups with Moore complex of length 2.
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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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.
DoldKan Type Theorem for ΓGroups
, 1998
"... Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan th ..."
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Introduction \Gammaspaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on KanThurston theorem we show that any \Gammaspace is stably weak equivalent to a discrete \Gammagroup. By a wellknown theorem of DoldKan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gammagroups. This construction is based on the notion of crosseffects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the nonabelian setup. Finally a DoldKan type theorem for the category of \Gammagroups is proved. In abelian case our theorem claims that the category of abelian \Gammagroups is equivalent to the category of functors Ab\Omega , where\Om
Freeness Conditions for Crossed Squares and Squared Complexes.
, 2008
"... Following Ellis, [9], we investigate the notion of totally free crossed square and related squared complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CWbasis, interms of the data for a totally free crossed square. Results of Ellis then apply to ..."
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Following Ellis, [9], we investigate the notion of totally free crossed square and related squared complexes. It is shown how to interpret the information in a free simplicial group given with a choice of CWbasis, interms of the data for a totally free crossed square. Results of Ellis then apply to give a description in terms of tensor products of crossed modules. The paper ends with a purely algebraic derivation of a result
Advances in Computing the Nonabelian Tensor Square of Polycyclic Groups
"... The nonabelian tensor square G ⊗ G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = ( g g ′ ⊗ g h)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h) ( h g ⊗ h h ′) for all g, g, h, h ′ ∈ G, where g g ′ = gg ′ g −1 is conjugation on the left. Following ..."
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The nonabelian tensor square G ⊗ G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg ′ ⊗ h = ( g g ′ ⊗ g h)(g ⊗ h) and g ⊗ hh ′ = (g ⊗ h) ( h g ⊗ h h ′) for all g, g, h, h ′ ∈ G, where g g ′ = gg ′ g −1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the more general nonabelian tensor product independently introduced by R. Brown and J.L. Loday [6]. By computing the nonabelian tensor square we mean finding a standard or simplified presentation for it. In the case of finite groups, the definition of the nonabelian tensor square gives a finite presentation that can be simplified using Tietze transformations. This simplified presentation can then be examined to determine the nonabelian tensor square. This was the approach taken in [3], in which the nonabelian tensor square was computed for each nonabelian group of order up to 30. Creating a presentation from the definition of the nonabelian tensor square, simplifying it using Tietze transformations and computing a structure description from the simplified presentation can be implemented in few lines of GAP [16]. However, this strategy does not scale well to finite groups G having order greater than 100 since the initial presentation has G  2 generators and 2G  3 relations. The most general method for computing the nonabelian tensor square uses the notion of a crossed pairing (see [3]). Let G and L be
This is a collation of the operative parts of a proposal submitted to the NSF concerning
"... cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themse ..."
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cal structures suggests, then sustained interaction is essential. Bringing people together will result in significant technical progress and significantly greater conceptual understanding of these structures. Despite their complexity, if developed coherently, these structures, like categories themselves, should eventually become part of the standard mathematical culture. PROJECT DESCRIPTION EXPLORATIONS OF HIGHER CATEGORICAL STRUCTURES AND THEIR APPLICATIONS PROPOSED RESEARCH 1. Historical background and higher homotopies Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper [47]. The language they introduced transformed modern mathematics. Their focus was not on categories and functors, but on natural transformations, which are maps between functors. Higher category theory concerns higher level notions of naturality, which can be viewed as maps between natural transformations, and maps between such maps, and so for
Project Description:
"... d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surf ..."
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d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through nmanifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an ncategory with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb
DOLDKAN TYPE THEOREMS FOR nTYPES OF SIMPLICITIAL COMMUTATIVE ALGEBRAS
, 1998
"... A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types. ..."
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A functor from simplicial algebras to crossed ncubes is shown to be an embedding on a reflexive subcategory of the category of simplicial algebras that contains representatives for all n types.