##  [Coalgebra](/coalgebra) 

  ##  [Coalgebra](https://algebra.quantumdictionary.io/coalgebra-0) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A coalgebra over a field or ring is a linear space C equipped with a comultiplication Δ: C→C⊗C and a counit ε: C→k satisfying coassociativity ((Δ⊗id)∘Δ=(id⊗Δ)∘Δ) and counitality ((ε⊗id)∘Δ=id=(id⊗ε)∘Δ). It is the categorical dual notion to an algebra when arrows are reversed.