Jirí Adámek

Infinite trees and completely iterative theories: a coalgebraic view

Infinite trees form a free completely iterative theory over any given signature--this fact, proved by Elgot, Bloom and Tindell, turns out to be a special case of a much more general categorical result exhibited in the present paper. We prove that whenever an endofunctor H of a category has final coalgebras for all functors H(-) + X, then those coalgebras, TX, form a monad. This monad is completely iterative, i.e., every guarded system of recursive equations has a unique solution. And it is a free completely iterative monad on H. The special case of polynomial endofunctors of the category Set is the above mentioned theory, or monad, of infinite trees.This procedure can be generalized to monoidal categories satisfying a mild side condition: if, for an object H, the endofunctor H ⊗ _ + I has a final coalgebra, T, then T is a monoid. This specializes to the above case for the monoidal category of all endofunctors.

On tree coalgebras and coalgebra presentations

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of this family of coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique At which takes the root of t to s. Consequently, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems.In contrast, for transition systems expressed as coalgebras over the finite-power-set functor we show that there are systems which fail to be filtered colimits of finitely presentable (=finite) ones.Surprisingly, if λ is an uncountable cardinal, then λ-presentation is always well-behaved: whenever an endofunctor F preserves λ-filtered colimits (i.e., is λ-accessible), then λ-presentable coalgebras are precisely those whose underlying objects are λ-presentable. Consequently, every F coalgebra is a λ-filtered colimit of λ-presentable coalgebras; thus Coalg F is a locally λ-presentable category. (This holds for all endofunctors of λ-accessible categories with colimits of ω-chains.) Corollary: A set functor is bounded at λ in: the sense of Kawahara and Mori iff it is λ+-accessible.

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

On final coalgebras of continuous functors

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Semantics of higher-order recursion schemes

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

A logic of coequations

Continuous endofunctors F of locally finitely presentable categories carry a natural metric on their final coalgebra. Whenever F(0) has an element, this metric is proved to be a Cauchy completion of the initial algebra of F. This is illustrated on the poset of real numbers represented as a final coalgebra of an endofunctor of Pos by Pavlovic´ and Pratt. Under additional assumptions on the locally finitely presentable category, all finitary endofunctors are proved to have a final coalgebra constructed in ω + ω steps of the natural iteration construction.

Semantics of Higher-Order Recursion Schemes

Higher-order recursion schemes are equations defining recursively new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of λ-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite λ-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Whereas Fiore et al proved that the presheaf Fλ of λ-terms is an initial Hλ-monoid, we work with the presheaf Rλ of rational infinite λ-terms and prove that this is an initial iterative Hλ-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in Rλ.

Syntactic Monoids in a Category

By Ruttens dualization of the Birkhoff Variety Theorem, a collection of coalgebras is a covariety (i.e., is closed under coproducts, subcoalgebras, and quotients) iff it can be presented by a subset of a cofree coalgebra. We introduce inference rules for these subsets, and prove that they are sound and complete. For example, given a polynomial endofunctor of a signature Σ, the cofree coalgebra consists of colored Σ-trees, and we prove that a set T of colored trees is a logical consequence of a set S iff T contains every tree such that all recolorings of all its subtrees lie in S. Finally, we characterize covarieties whose presentation needs only n colors.

On abstract data types presented by multiequations

Higher-order recursion schemes are recursive equations defining newnoperations from given ones called terminals. Every such recursion scheme isnproved to have a least interpreted semantics in every Scotts model ofnlambda-calculus in which the terminals are interpreted as continuousnoperations. For the uninterpreted semantics based on infinite lambda-terms wenfollow the idea of Fiore, Plotkin and Turi and work in the category of sets inncontext, which are presheaves on the category of finite sets. Fiore et alnshowed how to capture the type of variable binding in lambda-calculus by annendofunctor Hlambda and they explained simultaneous substitution ofnlambda-terms by proving that the presheaf of lambda-terms is an initialnHlambda-monoid. Here we work with the presheaf of rational infinitenlambda-terms and prove that this is an initial iterative Hlambda-monoid. Wenconclude that every guarded higher-order recursion scheme has a uniquenuninterpreted solution in this monoid.

On Final Coalgebras of Power-Set Functors and Saturated Trees To George Janelidze on the Occasion of His Sixtieth Birthday

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.