Algebras and Coalgebras

Algebras and coalgebras provide complementary ways to model construction and behavior.

An algebra consumes structure into a value:

F A -> A

A coalgebra unfolds a value into observable structure or next behavior:

A -> F A

Algebras

Algebras are useful for folds, accumulation, interpretation, and summarization.

Examples:

Folding committed events into state samples.
Accumulating observations into a projection.
Interpreting a command value into a transition decision.
Reducing validation results into accept, reject, or nil.

Coalgebras

Coalgebras are useful for behavior, observation, transition systems, and processes that expose next steps.

Examples:

A state machine exposing possible next transitions.
A process exposing its next required input, timeout, or command.
A behavior exposing current value and future evolution.
An observer exposing what it can observe, emit, or request next.

Algebras and coalgebras are connected to event-state duality. Folding events into state is algebraic. Observing state for possible events or future behavior is coalgebraic. Neither side fully replaces the other.