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 $\cal K$ is complete, exact, and has filtered colimits which (a) commute with finite limits and (b) distribute over products; besides (c) regular epimorphisms in $\cal K$ 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.
Theory and Applications of Categories, Vol. 8, 2001, No. 9, pp 253-283 http://www.tac.mta.ca/tac/volumes/8/n9/n9.dvi http://www.tac.mta.ca/tac/volumes/8/n9/n9.ps http://www.tac.mta.ca/tac/volumes/8/n9/n9.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n9/n9.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n9/n9.psTAC Home