 
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
