Date: Sat, 7 Dec 1996 16:37:57 -0400 (AST)
Subject: varieties

Date: Fri, 6 Dec 1996 14:25:38 +0100 (MET)
From: Jiri Rosicky <rosicky@math.muni.cz>

                 A DUALITY FOR VARIETIES OF ALGEBRAS
We have just proved a result we beleive is new - comments are highly
appreciated.
Theorem:
The following two 2-categories are dually biequivalent:
VAR - the category of all finitary, many-sorted varieties (O-cells)
      all finitary, regular right adjoints (1-cells)
      and all natural transformatons (2-cells)
CCFP - the 2-category of all Cauchy-complete, small categories
      with finite products (0-cells)
      all finite-products preserving functors (1-cells)

In particular, every variety has a "canonical" Lawvere theory
formed by all retracts of all finitely generated free algebras
of that variety (= of all finitely presentable projective objects).

J.Adamek & J.Rosicky