Hopf and Lie algebras in semi-additive Varieties

We study Hopf monoids in entropic semi-additive varieties with an emphasis on adjunctions related to the enveloping monoid functor and the primitive element functor. These investigations are based on the concept of the abelian core of a semi-additive variety variety and its monoidal structure in case the variety is entropic.Comment: 13 page

The history of the General Adjoint Functor Theorem

Not only motivated by the fact that the publication of the GAFT first appeared 60 years ago in print we reconstruct its history and so show that it is no exaggeration to claim that it has appeared already 75 years ago

Fuzzy functions: a fuzzy extension of the category SET and some related categories

[EN] In research Works where fuzzy sets are used, mostly certain usual functions are taken as morphisms. On the other hand, the aim of this paper is to fuzzify the concept of a function itself. Namely, a certain class of L-relations F : X x Y -> L is distinguished which could be considered as fuzzy functions from an L-valued set (X,Ex) to an L-valued set (Y,Ey). We study basic properties of these functions, consider some properties of the corresponding category of L-valued sets and fuzzy functions as well as briefly describe some categories related to algebra and topology with fuzzy functions in the role of morphisms.Höhle, U.; Porst, H.; Sostak, AP. (2000). Fuzzy functions: a fuzzy extension of the category SET and some related categories. Applied General Topology. 1(1):115-127. doi:10.4995/agt.2000.3028.SWORD1151271

On corings and comodules

It is shown that the categories of R-coalgebras for a commutative unital ring R and the category of A-corings for some R-algebra A as well as their respective categories of comodules are locally presentable

Universal Constructions for Hopf Algebras

AbstractThe category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are the various subcategory-embeddings arsing naturally in this context reflexive or coreflexive? (c) When does a category of Hopf monoids have all limits or colimits? These problems are also shown to be intimately related. Particular emphasis is given to the case of Hopf algebras, i.e., when the chosen symmetric monoidal category is the category of modules over a commutative unital ring