Universal Algebra v. Category Theory

19 Oct 1993

      Here are some remarks in connection with the Universal Algebra
versus Category Theory debate initiated by Saunders MacLane.

(i) There is yet another way to describe a (multisorted) algebraic
theory, alias clone, namely as a Cartesian multicategory, that is,
a Gentzen style deductive system with appropriate equations between
deductions.  Deductions f:A_1...A_n ---> B should be viewed as operations, in 
the spirit of the formulas as types paradigm.

(ii) Why anyone would deny that the subalgebras of an algebra form a complete 
lattice is beyond me.  The subobject lattice and congruence lattice are both 
subsumed in the lattice of subcongruences, which I found myself in my paper
"Goursat's theorem and the Zassenhaus lemma", Can. J. Math. 10(1957), 45-56.
Subcongruences have recently been resurrected as partial equivalence 
relations;
they can of course be defined in any category.

(iii) If one studies algebras on graphs instead of sets, as pioneered by 
Burroni, one may view many structured categories, even toposes, as algebras.

(iv) To avoid problems with empty types, e.g. lack of transitivity, one should 
declare variables both in equations and deductions. Phil Scott and I did so in 
our book of 1986, by placing a subscript X={x_1,...x_n} on the equality and
entailment symbols.  Unfortunately, some other books did not follow this 
practice.

Jim Lambek
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

es＠math.mcgill.ca

Joseph.Goguen＠prg.oxford.ac.uk

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