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Summary: ON ALGEBRAICALLY EXACT CATEGORIES AND
ESSENTIAL LOCALIZATIONS OF VARIETIES
J. Ad amek ) , J. Rosick y ) and E. M. Vitale
Abstract. Algebraically exact categories have been introduced in [ALR3 ] as an
equational hull of the 2-category VAR of all varieties of nitary algebras. We
will show that algebraically exact categories with a regular generator are precisely
the essential localizations of varieties and that, in this case, algebraic exactness is
equivalent to (1) exactness, (2) commutativity of ltered colimits with nite limits,
(3) distributivity of ltered colimits over arbitrary products and (4) product-stability
of regular epimorphisms. This can be viewed as a non-additive generalization of the
classical Roos Theorem characterizing essential localizations of categories of modules.
Analogously, precontinuous categories, introduced in [ALR2 ] as an equational hull of
the 2-category LFP (of locally nitely presentable categories) are characterized by
the above properties (2) and (3). Essential localizations of locally nitely presentable
categories and presheaf categories are fully described.
1. Introduction
The category
VAR
of ( nitary) varieties is not equational over CAT , the quasicategory of all categories,
as shown in [ALR 3 ]. There an equational hull of VAR with respect to \small
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