A Characterization of the Higgins Commutator

Abstract

We show that in an ideal-determined unital category the Higgins commutator can be characterized as the largest binary operation C on subobjects (defined on all subobjects of each object) satisfying the following conditions: (a) C is order-preserving; (b) C(H, K) is always less or equal to the meet of normal closures of H and K; (c) $$C(f(H),f(K))=f(C(H,K))$$ for every pair of subobjects H and K of an object X, and every morphism f whose domain is X.