fundamental isomorphism theorem for coalgebras

Let $(C,\Delta,\varepsilon)$ and $(D,\Delta^{\prime},\varepsilon^{\prime})$ be coalgebras. Recall, that if $D_{0}\subseteq D$ is a subcoalgebra, then $(D_{0},\Delta^{\prime}_{|D_{0}},\varepsilon^{\prime}_{|D_{0}})$ is a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then there is a canonical coalgebra structure on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details).

If $f:C\to D$ is a coalgebra homomorphism, then $\mathrm{ker}(f)$ is a coideal, $\mathrm{im}(f)$ is a subcoalgebra and a mapping $f^{\prime}:C/\mathrm{ker}(f)\to\mathrm{im}(f)$ defined by $f^{\prime}\big{(}c+\mathrm{ker}(f)\big{)}=f(c)$ is a well defined coalgebra isomorphism.

Title fundamental isomorphism theorem for coalgebras FundamentalIsomorphismTheoremForCoalgebras 2013-03-22 18:49:30 2013-03-22 18:49:30 joking (16130) joking (16130) 4 joking (16130) Theorem msc 16W30