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September 1994

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Top\op is a quasi-variety
by Michael Barr 02 Sep '94

02 Sep '94

by Michael Barr and M. Cristina Pedicchio We show that there is a certain variety (= category tripleable over sets) and a simple Horn sentence in it of the form phi(u) = phi(v) ==> psi(u) = psi(v) whose category of models is equivalent to the opposite of topological spaces. The theory consists of that of frames together with a unary operation we denote ' (it is a kind of pseudocomplement) satisfying a small set of equations plus an equation scheme that forces all intervals of the form [u /\ u',u \/ u'] to be complete atomic boolean algebras with the Sup and ' as operations. The underlying set functor on Top\op takes a space to the set of all pairs (U,A) where U is open and A is an arbitrary subset of U. The frame operations are the usual, while (U,A)' = (U,U - A). The Horn clause is u \/ u' \/ 1' = v \/ v' \/ 1' ==> u \/ u' = v \/ v'. [note from moderator: Michael says the paper will be available by ftp from triples.math.mcgill.ca soon.]

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