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Weak homomorphisms between functorial algebra

Schneider F. M.

Demonstratio Mathematica

2011

Vol. 44, nr 4

801-818

In universal algebra, homomorphisms are usually considered between algebras of the same similarity type. Different from that, the notion of a weak homomorphism, as introduced by E. Marczewski in 1961, does not depend on a signature, but only on the clones of term operations generated by the examined algebras. We generalize this idea by defining weak homomorphisms between F1 - and F2-algebras, where F1 and F2 denote not necessarily equal endofunctors of the category of sets. The aim is to show that, in many respects, weak homomorphisms behave very similarly to proper homomorphisms-without restricting the scope of considerations by the necessity of a common type. For instance, concerning a set F of Set -endofunctors that weakly preserve kernels, the class of all algebras of types from F equipped with the class of all weak homomorphisms between these algebras forms a category which admits a canonical factorization structure for morphisms. Furthermore, we treat two product constructions from which the notion of a weak homomorphism naturally arises.