31 research outputs found
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
Algebraic theories of quasivarieties
Analogously to the fact that Lawvereâs algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of manyâsorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebraic theories of manyâsorted Horn classes are precisely the small left exact categories with enough Mâinjectives, where M is a class of monomorphisms closed under finite products and containing all regular monomorphisms. We also present a GabrielâUlmerâtype duality theory for quasivarieties and Horn classes. 1 Quasivarieties and Horn Classes The aim of the present paper is to describe, via algebraic theories, classes of finitary algebras, or finitary structures, which are presentable by implications. We work with finitary manyâsorted algebras and structures, but we also mention the restricted version to the oneâsorted case on the one hand, and the generalization to infinitary structures on the other hand. Recall that Lawvereâs thesis [11] states that Lawvereâtheories of varieties, i.e., classes of algebras presented by equations, are precisely the small categories with finite products, (in the one sorted case moreover productâgenerated by a single object; for manyâsorted varieties the analogous statement can be found in [4, 3.16, 3.17]). More in detail: If we denote, for small categories A, by P rodÏA the full subcategory of Set A formed by all functors preserving finite products, we obtain the following: (i) If K is a variety, then its Lawvereâtheory L(K), which is the full subcategory of K op of all finitely generated free Kâalgebras, is essentially small, and has finite products. The variety K is equivalent to P rodÏL(K)
On Categories of Monoids, Comonoids, and
For the sixtieth birthday of my friend and colleague JiËrĂ AdĂĄmek The categories of monoids, comonoids and bimonoids over a symmetric monoidal category C are investigated. It is shown that all of them are locally presentable provided Câs underlying category is. As a consequence numerous functors on and between these categories ar