# How algebraic is algebra?

## J. Adamek, F. W. Lawvere and J. Rosicky

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.

Keywords: variety, exact category, pseudomonad.

2000 MSC: 18C99, 18D99, 08B99.

Theory and Applications of Categories, Vol. 8, 2001, No. 9, pp 253-283.

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