Jiri Velebil

Iterative algebras at work

Iterative theories, which were introduced by Calvin Elgot, formalise potentially infinite computations as unique solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely complicated. In our paper we show that by starting with ‘iterative algebras’, that is, algebras admitting a unique solution of all systems of flat recursive equations, a free iterative theory is obtained as the theory of free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic one. Despite this, our result is much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category

Equational presentations of functors and monads

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Extensions of Functors From Set to V-cat

.Reportedly, a blow from the welterweight boxer Norman Selby, also known as Kid McCoy, left one victim proclaiming,‘Its the real McCoy!’.

Final Coalgebras in Accessible Categories

We study equational presentations of functors and monads defined on a category that is equipped by an adjunction F ?? U : ??? of descent type. We present a class of functors/monads that admit such an equational presentation that involves finitary signatures in . We apply these results to an equational description of functors arising in various areas of theoretical computer science.

Categorical Generalization of a Universal Domain

We show that for a commutative quantale V every functor from Set to V-cat has an enriched left-Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.

Semantics of higher-order recursion schemes

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 self-similar objects in topology.

Expressiveness of Positive Coalgebraic Logic.

J. Adámek defined SC categories as a categorical generalization of Scott domains. Namely, an SC category is finitely accessible, has an initial object and is boundedly cocomplete (each diagram with a compatible cocone has a colimit). SC categories are proved to serve well as a basis for the computer language semantics.The purpose of this paper is to generalize the concept of a universal Scott domain to a universal SC category. We axiomatize properties of subcategories of finitely presentable objects of SC categories (generalizing thus semilattices of compact elements of Scott domains). The categories arising are called FCC (finitely consistently cocomplete) categories. It is shown that there exists a universal FCC category, i.e., such that every FCC category may be FCC embedded into it. The result is an application of a general procedure introduced 30 years ago by V. Trnková.