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On iterable endofunctors
 Category Theory and Computer Science 2002, number 69 in Elect. Notes in Theor. Comp. Sci
, 2003
"... Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H ( ) + X. We show that conversely, if ..."
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Completely iterative monads of Elgot et al. are the monads such that every guarded iterative equation has a unique solution. Free completely iterative monads are known to exist on every iteratable endofunctor H, i. e., one with final coalgebras of all functors H ( ) + X. We show that conversely, if H generates a free completely iterative monad, then it is iteratable. Key words: monad, completely iterative, iterable 1
Codes and Equations on Trees
, 1998
"... The objective of this paper is to study, by new formal methods, the notion of tree code introduced by M. Nivat in [23]. In particular we introduce the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the ..."
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The objective of this paper is to study, by new formal methods, the notion of tree code introduced by M. Nivat in [23]. In particular we introduce the notion of stability for sets of trees closed under concatenation. This allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids. We further define the stable hull of a set of trees and derive a defect theorem for trees, which generalizes the analogous result for words. As a consequence we obtain some properties of tree codes having two elements. Moreover we propose a new algorithm to test whether a finite set of trees is a tree code. The running time of the algorithm is polynomial in the size of the input. We also introduce the notion of tree equation as a complementary point of view to tree codes. The main problem emerging in this approach is to decide the satisfiability of tree equations: it is a special case of second order unification, and it remains still open.
FINAL COALGEBRAS IN ACCESSIBLE CATEGORIES
, 905
"... Abstract. We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in ..."
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Abstract. We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of selfsimilar objects in topology. 1.