In mathematics, specifically in category theory, an ${\displaystyle F}$-coalgebra is a structure defined according to a functor ${\displaystyle F}$, with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy evaluation, infinite data structures, such as streams, and also transition systems.

${\displaystyle F}$-coalgebras are dual to ${\displaystyle F}$-algebras. Just as the class of all algebras for a given signature and equational theory form a variety, so does the class of all ${\displaystyle F}$-coalgebras satisfying a given equational theory form a covariety, where the signature is given by ${\displaystyle F}$.

Let

be an endofunctor on a category ${\displaystyle {\mathcal {C}}}$. An ${\displaystyle F}$-coalgebra is an object ${\displaystyle A}$ of ${\displaystyle {\mathcal {C}}}$ together with a morphism

of ${\displaystyle {\mathcal {C}}}$, usually written as ${\displaystyle (A,\alpha )}$.

An ${\displaystyle F}$-coalgebra homomorphism from ${\displaystyle (A,\alpha )}$ to another ${\displaystyle F}$-coalgebra ${\displaystyle (B,\beta )}$ is a morphism

in ${\displaystyle {\mathcal {C}}}$ such that

Thus the ${\displaystyle F}$-coalgebras for a given functor F constitute a category.