Coalgebra

Summary

A coalgebra is a vector space $C$ over a field $K$ with a comultiplication $\Delta \colon C \to C \otimes C$ and a counit $\varepsilon \colon C \to K$. Its axioms come from reversing the arrows in the definition of an associative algebra. The dual space of a coalgebra is an algebra, and the dual of a finite-dimensional algebra is a coalgebra. Coalgebras are the other half of bialgebras and Hopf algebras, and they model ways of splitting an object into two pieces.