Combinatorial homotopy. I

Whitehead, J. H. C.

doi:10.1090/s0002-9904-1949-09175-9

Cited by 521 publications

(185 citation statements)

References 21 publications

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“…Two homotopy equivalent spaces are weakly homotopy equivalent (the converse is not true in general but the Whitehead's theorem [39,40]) states that it is true in complexes (see Section 4.1).…”

Section: Algebraic Topologymentioning

confidence: 99%

Paths, Homotopy and Reduction in Digital Images

et al. 2010

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The development of digital imaging (and its subsequent applications) has led to consider and investigate topological notions, well-defined in continuous spaces, but not necessarily in discrete/digital ones. In this article, we focus on the classical notion of path. We establish in particular that the standard definition of path in algebraic topology is coherent w.r.t. the ones (often empirically) used in digital imaging. From this statement, we retrieve, and actually extend, an important result related to homotopy-type preservation, namely the equivalence between the fundamental group of a digital space and the group induced by digital paths. Based on this sound definition of paths, we also (re)explore various (and sometimes equivalent) ways to reduce a digital image in a homotopy-type preserving fashion.

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Section: Algebraic Topologymentioning

confidence: 99%

Paths, Homotopy and Reduction in Digital Images

et al. 2010

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“…Clearly, for the latter structure this fact is true by definition (for any spaces). As for the former, the corresponding fact simply restates the following well known theorem of J.H.C.Whitehead [28,29] (note that CW -complexes are cofibrant and fibrant in the first structure).…”

Section: Introductionmentioning

confidence: 71%

Topological Model Categories Generated by Finite Complexes

Chigogidze

Karasev

2003

Monatshefte f�r Mathematik

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Abstract. Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -is obtained by letting L = {point}. The other -describing n-homotopy equivalences between at most (n + 1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ≤ n -by letting L = S n+1 , n ≥ 0.

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“…Eilenberg and Mac Lane [Eilenberg and Mac Lane 1953;1954;Mac Lane 1952] argue that the cohomology groups H n (A, ‫ދ‬ × ) are inappropriate since they do not take into account the abelianness of A, and so should be replaced by groups H n ab (A, ‫ދ‬ × ). (For the cohomology theory for crossed modules, see [Whitehead 1949]. ) Below we recall the definition of H 3 ab (A, ‫ދ‬ × ).…”

Section: Quasiabelian Third Cohomology Of Crossed Modulesmentioning

confidence: 99%

Crossed pointed categories and their equivariantizations

Naidu¹

2010

Pacific J. Math.

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We propose a notion, quasiabelian third cohomology of crossed modules, which generalizes Eilenberg and Mac Lane's abelian and Ospel's quasiabelian cohomology. We classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories, which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. We give a criterion for these categories to be modular. We describe the quasitriangular quasi-Hopf algebras underlying these categories. IntroductionTuraev's notion [2000; 2008] of a crossed category (short for braided group-crossed category) has attracted much attention recently [Drinfeld et al. 2010;Kirillov 2001a;2001b;Müger 2004;. Roughly, a crossed category consists of a group G, a G-graded tensor category Ꮿ, an action g → T g of G on Ꮿ by tensor autoequivalences, and G-braidings c(X, Y ) : X ⊗ Y ∼ − → T g (Y ) ⊗ X for X, Y ∈ Ꮿ, satisfying certain compatibility conditions. Crossed categories are known to arise in various contexts; for instance, Müger [2004] showed that Galois extensions of braided tensor categories have a natural structure of crossed categories. In [2005], Müger established a connection between 1-dimensional quantum field theories and crossed categories. Kirillov [2001b] showed that crossed categories arise in the theory of vertex operator algebras.A fusion category is said to be pointed if all its simple objects are invertible. One of the goals of this paper is to classify all crossed pointed categories. From [Joyal and Street 1993], it is known that braided pointed categories are classified by Eilenberg and Mac Lane's abelian cohomology H 3 ab (A, ‫ދ‬ × ), where A is a finite abelian group. On the other hand, certain crossed pointed categories in which the group action is strict were described by Turaev [2000;2008] in terms of Ospel's quasiabelian cohomology H 3 qa (G, ‫ދ‬ × ), where G is a (not necessarily abelian) finite MSC2000: primary 18D10; secondary 16W30.

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Combinatorial homotopy. I

Cited by 521 publications

References 21 publications

Paths, Homotopy and Reduction in Digital Images

Paths, Homotopy and Reduction in Digital Images

Topological Model Categories Generated by Finite Complexes

Crossed pointed categories and their equivariantizations

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