F-coalgebra

In mathematics, specifically in category theory, an $F$ -coalgebra for an endofunctor

$F : \mathbf{C}\longrightarrow \mathbf{C}$

is an object $A$ of $\mathbf{C}$ together with a $\mathbf{C}$ -morphism

$\alpha : A \longrightarrow FA$ .

In this sense F-coalgebras are dual to F-algebras.

Homomorphisms of $F$ -coalgebras are morphisms

$f:A\longrightarrow B$

in $\mathbf{C}$ such that

$Ff\circ \alpha = \beta \circ f$ .

Thus $F$ -coalgebras for a given functor F constitute a category.

Examples

Consider the functor $F: \mathbf{Set} \longrightarrow \mathbf{Set}$ that sends $X$ to $X + 1$ , $F$ -coalgebras $\alpha : X \longrightarrow X+1 = FX$ are then "streams", where the elements of $X$ are the states and $α$ is the transition to the next state and 1 means "end of file".

F-coalgebra

Examples

See also: F-coalgebra, Category theory, Coalgebra, Homomorphism, Mathematics, Morphism, F-algebra, Endofunctor