## Monads from inductive and coinductive types

Institute of Cybernetics

Tuesday, 12 March 2002, 15:00

Cybernetica Bldg (Akadeemia tee 21), room B216

**Abstract**: It is well known that, given an endofunctor
*H* on a category *C*, the initial *(A + H -)*-algebras
for different objects A of *C* (if they all exist), i.e., the
algebras of wellfounded H-branching trees-with-variables over
different variable supplies A, give rise to a monad with substitution
as the extension operation. A similar monad, but with the additional
property of being "completely iterative", is induced by the inverses
of the final *(A + H -)*-coalgebras (if they exist), i.e., the
algebras of non-wellfounded H-branching trees-with-variables. Aczel,
Adámek, Milius, Velebil (2001) have given one very neat
generalization of these facts. We consider a different one: if
*T'* is a bifunctor such that the functors *T'(-, X)*
uniformly carry a monad structure, then the initial *T'(A,
-)*-algebras (if existing) give rise to a monad and the inverses of
the final *T'(A, -)*-coalgebras (if existing) yield an
"iterative" monad.

Tarmo Uustalu