coalgebra homomorphism

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

Definition. Linear map $f:C\\to D$ is called coalgebra homomorphism if $\\Delta^{\\prime}\\circ f=(f\\otimes f)\\circ\\Delta$ and $\\varepsilon^{\\prime}\\circ f=\\varepsilon$ .

Examples. $1)$ Of course, if $D$ is a subcoalgebra of $C$ , then the inclusion $i:D\\to C$ is a coalgebra homomorphism. In particular, the identity is a coalgebra homomorphism.

$2)$ If $(C,\\Delta,\\varepsilon)$ is a coalgebra and $I\\subseteq C$ is a coideal, then we have canonical coalgebra structur on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details). Then the projection $\\pi:C\\to C/I$ is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra structure on $C/I$ is a unique coalgebra structure such that $\\pi$ is a coalgebra homomorphism.

单词	CoalgebraHomomorphism
释义	coalgebra homomorphism Let $(C,\\Delta,\\varepsilon)$ and $(D,\\Delta^{\\prime},\\varepsilon^{\\prime})$ be coalgebras. Definition. Linear map $f:C\\to D$ is called coalgebra homomorphism if $\\Delta^{\\prime}\\circ f=(f\\otimes f)\\circ\\Delta$ and $\\varepsilon^{\\prime}\\circ f=\\varepsilon$ . Examples. $1)$ Of course, if $D$ is a subcoalgebra of $C$ , then the inclusion $i:D\\to C$ is a coalgebra homomorphism. In particular, the identity is a coalgebra homomorphism. $2)$ If $(C,\\Delta,\\varepsilon)$ is a coalgebra and $I\\subseteq C$ is a coideal, then we have canonical coalgebra structur on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details). Then the projection $\\pi:C\\to C/I$ is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra structure on $C/I$ is a unique coalgebra structure such that $\\pi$ is a coalgebra homomorphism.
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