Algebras, Coalgebras, Monads and Comonads

Neil Ghani, Christoph Lüth, Federico de Marchi, John Power

Research output: Contribution to journalArticlepeer-review

21 Citations (SciVal)


Whilst the relationship between initial algebras and monads is well-understood, the relationship between final coalgebras and previous termcomonadsnext term is less well explored. This paper shows that the problem is more subtle and that final coalgebras can just as easily form monads as previous termcomonads and dually, that initial algebras form both monads and previous termcomonads.

In developing these theories we strive to provide them with an associated notion of syntax. In the case of initial algebras and monads this corresponds to the standard notion of algebraic theories consisting of signatures and equations: models of such algebraic theories are precisely the algebras of the representing monad. We attempt to emulate this result for the coalgebraic case by defining a notion cosignature and coequation and then proving the models of this syntax are precisely the coalgebras of the representing comonad.

Original languageEnglish
Pages (from-to)128-145
Number of pages18
JournalElectronic Notes in Theoretical Computer Science
Issue number1
Publication statusPublished - 2001


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