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Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems. (English) Zbl 1067.18004
Janelidze, George (ed.) et al., Galois theory, Hopf algebras, and semiabelian categories. Papers from the workshop on categorical structures for descent and Galois theory, Hopf algebras, and semiabelian categories, Toronto, ON, Canada, September 23–28, 2002. (ISBN 0-8218-3290-5/hbk). Fields Institute Communications 43, 101-130 (2004).
Summary: We outline the main features of the definitions and applications of crossed complexes and cubical $$\omega$$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.
For the entire collection see [Zbl 1051.18002].

##### MSC:
 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 16E05 Syzygies, resolutions, complexes in associative algebras 18D35 Structured objects in a category (MSC2010) 55P15 Classification of homotopy type 55Q05 Homotopy groups, general; sets of homotopy classes
XMod; Gpd
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