coalgebra isomorphisms and isomorphic coalgebras

Let (C,Δ,ε) and (D,Δ,ε) be coalgebras.

Definition. We will say that coalgebra homomorphism f:CD is a coalgebra isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, if there exists a coalgebra homomorphism g:DC such that fg=idD and gf=idC.

Remark. Of course every coalgebra isomorphism is a linear isomorphism, thus it is ,,one-to-one” and ,,onto”. One can show that the converseMathworldPlanetmath also holds, i.e. if f:CD is a coalgebra homomorphism such that f is ,,one-to-one” and ,,onto”, then f is a coalgebra isomorphism.

Definition. We will say that coalgebras (C,Δ,ε) and (D,Δ,ε) are isomorphic if there exists coalgebra isomorphism f:CD. In this case we often write (C,Δ,ε)(D,Δ,ε) or simply CD if structure mapsPlanetmathPlanetmathPlanetmath are known from the context.

Remarks. Of course the relationMathworldPlanetmathPlanetmath ,,” is an equivalence relationMathworldPlanetmath. Furthermore, (from the coalgebraic point of view) isomorphic coalgebras are the same, i.e. they share all coalgebraic properties.

Title coalgebra isomorphisms and isomorphic coalgebras
Canonical name CoalgebraIsomorphismsAndIsomorphicCoalgebras
Date of creation 2013-03-22 18:49:28
Last modified on 2013-03-22 18:49:28
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 16W30