coalgebra isomorphisms and isomorphic coalgebras

Let $(C,\mathrm{\Delta },\epsilon )$ and $(D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ be coalgebras.

Definition. We will say that coalgebra homomorphism $f:C\to D$ is a coalgebra isomorphism, if there exists a coalgebra homomorphism $g:D\to C$ such that $f\circ g={\mathrm{id}}_D$ and $g\circ f={\mathrm{id}}_C$.

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 $f:C\to D$ 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 $(C,\mathrm{\Delta },\epsilon )$ and $(D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ are isomorphic if there exists coalgebra isomorphism $f:C\to D$. In this case we often write $(C,\mathrm{\Delta },\epsilon )\simeq (D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ or simply $C\simeq D$ if structure maps are known from the context.

Remarks. Of course the relation ,,$\simeq$” 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
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