coalgebra isomorphisms and isomorphic coalgebras
Let and be coalgebras.
Definition. We will say that coalgebra homomorphism is a coalgebra isomorphism, if there exists a coalgebra homomorphism such that and .
Remark. Of course every coalgebra isomorphism is a linear isomorphism, thus it is ,,one-to-one” and ,,onto”. One can show that the converse also holds, i.e. if is a coalgebra homomorphism such that is ,,one-to-one” and ,,onto”, then is a coalgebra isomorphism.
Definition. We will say that coalgebras and are isomorphic if there exists coalgebra isomorphism . In this case we often write or simply if structure maps are known from the context.
Remarks. Of course the relation ,,” is an equivalence relation
. 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 |
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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 |