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How algebraic is algebra?

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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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