This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabelian categories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers, specifically those interested in Galois theory and Hopf algebras and their categorical unification.
Algebraic cohomology: The early days by M. Barr A survey of semi-abelian categories by F. Borceux Commutator theory in regular Mal'cev categories by D. Bourn Categorical aspects of modularity by D. Bourn and M. Gran Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems by R. Brown Galois groupoids and covering morphisms in topos theory by M. Bunge Galois corings from the descent theory point of view by S. Caenepeel Quantum categories, star autonomy, and quantum groupoids by B. Day and R. Street Morphisms of 2-groupoids and low-dimensional cohomology of crossed modules by J. W. Duskin, R. W. Kieboom, and E. M. Vitale Applications of categorical Galois theory in universal algebra by M. Gran Fibrations for abstract multicategories by C. Hermida Lie-Rinehart algebras, descent, and quantization by J. Huebschmann A note on the semiabelian variety of Heyting semilattices by P. Johnstone Monoidal functors generated by adjunctions, with applications to transport of structure by G. M. Kelly and S. Lack On the cyclic homology of Hopf crossed products by M. Khalkhali and B. Rangipour On sequentially $h$-complete groups by G. Lukacs Embeddings of algebras by J. L. MacDonald Universal covers and category theory in polynomial and differential Galois theory by A. R. Magid Weak categories in additive 2-categories with kernels by N. Martins-Ferreira Dendrotopic sets by T. Palm On factorization systems and admissible Galois structures by A. H. Roque Hopf-Galois and bi-Galois extensions by P. Schauenburg Extension theory in Mal'tsev varieties by J. D. H. Smith On projective generators relative to coreflective classes by L. Sousa The monotone-light factorization for categories via preorders by J. J. Xarez Separable morphisms of categories via preordered sets by J. J. Xarez Frobenius algebras in tensor categories and bimodule extensions by S. Yamagami.