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How algebraic is algebra?
Theory and applications of categories, Tome 8 (2001), pp. 253-283.

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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.
Classification : 18C99, 18D99, 08B99.
Keywords: variety, exact category, pseudomonad. 
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     author = {J. Adamek and F. W. Lawvere and J. Rosicky},
     title = {How algebraic is algebra?},
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     pages = {253--283},
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     volume = {8},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2001_8_a8/}
}
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J. Adamek; F. W. Lawvere; J. Rosicky. How algebraic is algebra?. Theory and applications of categories, Tome 8 (2001), pp. 253-283. http://geodesic.mathdoc.fr/item/TAC_2001_8_a8/