Electronic Notes in Theoretical Computer Science 44 No. 1 (2001) URL: http://www.elsevier.nl/locate/entcs/volume44.html 26 pages
A Coalgebraic View of Infinite Trees and Iteration
Peter Aczel 1
Department of Mathematics and Computer Science, Manchester University, Manchester, United Kingdom
Jiˇr´ı Ad´amek 2,4
Institute of Theoretical Computer Science, Technical University, Braunschweig, Germany
Jiˇr´ı Velebil 3,4
Faculty of Electrical Engineering, Technical University, Praha, Czech Republic
Abstract The algebra of infinite trees is, as proved by C. Elgot, completely iterative, i.e., all ideal recursive equations are uniquely solvable. This is proved here to be a general coalgebraic phenomenon: let H be an endofunctor such that for every object X afi- nal coalgebra, TX,ofH( )+X exists. Then TX is an object-part of a monad which is completely iterative. Moreover, a similar contruction of a “completely iterative monoid” is possible in every monoidal category satisfying mild side conditions. Key words: monad, coalgebra, monoidal category
1 Email:[email protected] 2 Email:[email protected] 3 Email:[email protected] 4 The support of the Grant Agency of the Czech Republic under the Grant No. 201/99/0310 is gratefully acknowledged.