Algebras and Coalgebras

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

An algebra consumes structure into a value:

$$F A\to A$$

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

$$A\to FA$$

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.

Related concepts: behavior, event-state duality, event, state, transition, observer, process graphs, duality and symmetry.