coalgebra homomorphism

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

Definition. Linear map f:CD is called coalgebra homomorphism if Δf=(ff)Δ and εf=ε.

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

2) If (C,Δ,ε) is a coalgebra and IC is a coideal, then we have canonical coalgebra structur on C/I (please, see this entry ( for more details). Then the projection π:CC/I is a coalgebra homomorphism. Furthermore, one can show that the canonical coalgebra structureMathworldPlanetmath on C/I is a unique coalgebra structure such that π is a coalgebra homomorphism.

Title coalgebra homomorphism
Canonical name CoalgebraHomomorphism
Date of creation 2013-03-22 18:49:25
Last modified on 2013-03-22 18:49:25
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 16W30