How algebraic is algebra?
Jiřı́ Adámek, F. William Lawvere, Jiřı́ Rosický · Theory and applications of categories · 2001
. The 2-category VAR of finitary varieties is not varietal over CAT . We introduce the concept of an algebraically exact category and prove that the 2-category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations 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 product-stable. It is not known whether (a) -- (c) characterize algebraic exactness. An equational hull of VAR w.r.t. all operations is also discussed. 1.