Jiří Velebil

On coalgebra based on classes

The category Class of classes and functions is proved to have a number of properties suitable for algebra and coalgebra: every endofunctor is set-based, it has an initial algebra and a terminal coalgebra, the categories of algebras and coalgebras are complete and cocomplete, and every endofunctor generates a free completely iterative monad. A description of a terminal coalgebra for the power-set functor is provided.

A Coalgebraic View of Infinite Trees and Iteration

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 a final coalgebra, TX, of H(_) + 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.

On Rational Monads and Free Iterative Theories

Abstract For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R is a free iterative theory on H for every finitary endofunctor H, which is substantially simpler than the previous proof by Elgot et al., as well as our previous proof. This result holds for more general categories than Set.

Enriched Logical Connections

In the setting of enriched category theory, we describe dual adjunctions of the form

From Iterative Algebras to Iterative Theories

L\dashv R:{\mathsf{Spa}}^{op} \longrightarrow{\mathsf{Alg}}

What are iteration theories

between the dual of the category Spa of “spaces” and the category Alg of “algebras” that arise from a schizophrenic object Ω, which is both an “algebra” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of Ω is that of a module that allows to lift optimally the structure of a “space” and an “algebra” to certain diagrams. Our approach allows to give a unified framework known from logical connections over the category of sets and analyzed, e.g., by Hans Porst and Walter Tholen, with future applications of logical connections in coalgebraic logic and elsewhere, where typically, both the category of “spaces” and the category of “algebras” consist of “structured presheaves”.

Positive Fragments of Coalgebraic Logics

Iterative theories introduced by Calvin Elgot formalize potentially infinite computations as 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”, i. e., 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. And our result is, nevertheless, much more general: we describe a free iterative theory on any finitary endofunctor of every locally presentable category A. This allows us, e. g., to consider iterative algebras over any equationally specified class A of finitary algebras.

Iterative reflections of monads

We prove that iteration theories can be introduced as algebras for the monad Rat on the category of signatures assigning to every signature Σ the rational-Σ-tree signature. This supports the result that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: Rat is the monad of free rational theories and every free rational theory has a continuous completion.

A general final coalgebra theorem

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond.

Final Coalgebras And a Solution Theorem for Arbitrary Endofunctors

Iterative monads were introduced by Calvin Elgot in the 1970s and are those ideal monads in which every guarded system of recursive equations has a unique solution. We prove that every ideal monad has an iterative reflection, that is, an embedding into an iterative monad with the expected universal property. We also introduce the concept of iterativity for algebras for the monad , following in the footsteps of Evelyn Nelson and Jerzy Tiuryn, and prove that is iterative if and only if all free algebras for are iterative algebras.