Mathematical Structures in Computer Science

Dualising initial algebras

a1 Department of Mathematics and Computer Science, University of Leicester
a2 FB 3 – Mathematik und Informatik, Universität Bremen
a3 Laboratory for Foundations of Computer Science, University of Edinburgh


Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads.

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 first defining a notion of cosignature and coequation and then proving that the models of such coalgebraic presentations are precisely the coalgebras of the representing comonad.

(Received August 30 2001)
(Revised March 18 2002)


1 Research supported by EPSRC grant GRM96230/01: Categorical Rewriting: Monads and Modularity.