A Categorical Approach to Turaev’s Hopf Group-coalgebras

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known...

Hopf-Algebras and Coalgebras in ∞-Categories

The idea of the following is to suggest an ∞-categorical framework within which the ideas of coalgebras and Hopf-algebras (for various degrees of commutativity and cocommutativity) are natural and easy to define. These foundations will be used to describe interesting examples arising in chromatic homotopy theory and elsewhere in forthcoming work (especially with respect to Thom spectra). Sectio...

Covering Coalgebras and Dual Non-singularity

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843– 2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coal...

Topology on Coalgebras

Coalgebras, Hopf Algebras and Combinatorics

In loving memory of my mother " A mathematician is a machine for converting coffee into theorems. " —Alfréd Rényi " A comathematician, by categorical duality, is a machine for converting cotheorems into ffee. " —anonymous Preface Hopf algebras are a relatively new concept in algebra, first encountered by their namesake Heinz Hopf in 1941 in the field of algebraic topology. In the 1960s, study o...