Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-28T11:46:55.871Z Has data issue: false hasContentIssue false

Dualising initial algebras

Published online by Cambridge University Press:  01 April 2003

NEIL GHANI
Affiliation:
Department of Mathematics and Computer Science, University of Leicester
CHRISTOPH LÜTH
Affiliation:
FB 3 – Mathematik und Informatik, Universität Bremen
FEDERICO DE MARCHI
Affiliation:
Department of Mathematics and Computer Science, University of Leicester
JOHN POWER
Affiliation:
Laboratory for Foundations of Computer Science, University of Edinburgh

Abstract

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.

Type
Research Article
Copyright
2003 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)