Summary: HOW ALGEBRAIC IS ALGEBRA ?
J. AD '
AMEK \Lambda) , F. W. LAWVERE AND J. ROSICK '
Y \Lambda)
ABSTRACT. The 2­category VAR of finitary varieties is not varietal over CAT . We
introduce the concept of an algebraically exact category and prove that the 2­category
ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all op­
erations with rank. Every algebraically exact category K is complete, exact, and has
filtered colimits which (a) commute with finite limits and (b) distribute over products;
besides (c) regular epimorphisms in K are product­stable. It is not known whether (a)
-- (c) characterize algebraic exactness. An equational hull of VAR w.r.t. all operations
is also discussed.
I. Introduction
I.1 Is algebra algebraic? The purpose of our paper is to study the non­full embed­
ding
U : VAR ! CAT
of the 2­category of all finitary varieties into the 2­quasicategory of all categories. The
morphisms (1­cells) of the former are indicated by the duality between varieties and alge­
braic theories introduced in [ALR 1 ]: they are the algebraically exact functors, i.e., finitary
right adjoints preserving regular epimorphisms. And 2­cells are the natural transforma­

Source: Adámek, Jiri - Institut für Theoretische Informatik, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig

Collections: Computer Technologies and Information Sciences