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Weak homomorphisms of coalgebras beyond Set

Kianpi M.

Demonstratio Mathematica

2014

Vol. 47, nr 3

555--580

We study the notion of weak homomorphisms between coalgebras of different types generalizing thereby that of homomorphisms for similarly typed coalgebras. This helps extend some results known so far in the theory of Universal coalgebra over Set. We find conditions under which coalgebras of a set of types and weak homomorphisms between them form a category. Moreover, we establish an Isomorphism Theorem that extends the so-called First Isomorphism Theorem, showing thereby that this category admits a canonical factorization structure for morphisms.

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.