Terminal Coalgebra Research Articles

Homotopy Theories of (∞, ∞)-Categories as Universal Fixed Points With Respect to Weak Enrichment

Abstract We show that both the $\infty $-category of $(\infty , \infty )$-categories with inductively defined equivalences, and with coinductively defined equivalences, satisfy universal properties with respect to weak enrichment in the sense of Gepner and Haugseng. In particular, we prove that $(\infty , \infty )$-categories with coinductive equivalences form a terminal object in the $\infty $-category of fixed points for enrichment, and that $(\infty , \infty )$-categories with inductive equivalences form an initial object in the subcategory of locally presentable fixed points. To do so, we develop an analogue of Adámek’s construction of free endofunctor algebras in the $\infty $-categorical setting. We prove that $(\infty , \infty )$-categories with coinductive equivalences form a terminal coalgebra with respect to weak enrichment, and $(\infty , \infty )$-categories with inductive equivalences form an initial algebra with respect to weak enrichment.

Grounding Game Semantics in Categorical Algebra

I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.

Fixed points of functors

This is a survey on fixed points of endofunctors, including initial algebras and terminal coalgebras. We also consider the rational fixed point, a canonical domain of behavior for finitely presentable systems. In addition to the basic existence theorems for fixed points, several new results are presented. For example, the Smyth–Plotkin theorem that locally continuous endofunctors of DCPO have terminal coalgebras is derived from a new result stating that every locally monotone endofunctor with a fixed point has a terminal coalgebra. We introduce bounded endofunctors on abstract categories and prove that they have terminal coalgebras. We study well-founded coalgebras and prove that for set functors, the largest well-founded coalgebra of every fixed point is the initial algebra. Another new result concerns mixed fixed points: initial algebras and terminal coalgebras of a parametrized accessible functor always form accessible functors.

Fixed Points of Set Functors: How Many Iterations are Needed?

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)

Corecursive Algebras, Corecursive Monads and Bloom Monads

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras.

Abstract GSOS Rules and a Modular Treatment of Recursive Definitions

Terminal coalgebras for a functor serve as semantic domains for state-based systems of various types. For example, behaviors of CCS processes, streams, infinite trees, formal languages and non-well-founded sets form terminal coalgebras. We present a uniform account of the semantics of recursive definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS rules l specify additional algebraic operations on a terminal coalgebra; (2) terminal coalgebras are also initial completely iterative algebras (cias). We also show that an abstract GSOS rule leads to new extended cia structures on the terminal coalgebra. Then we formalize recursive function definitions involving given operations specified by l as recursive program schemes for l, and we prove that unique solutions exist in the extended cias. From our results it follows that the solutions of recursive (function) definitions in terminal coalgebras may be used in subsequent recursive definitions which still have unique solutions. We call this principle modularity. We illustrate our results by the five concrete terminal coalgebras mentioned above, e.\,g., a finite stream circuit defines a unique stream function.

Relatively terminal coalgebras

Dana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F. We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-to-algebra morphism into A.We prove that by iterating F countably many times we obtain the relatively terminal coalgebras whenever F preserves limits of ωop-chains. If F is finitary, we need in general ω+ω steps. And for arbitrary accessible (=bounded) set functors we need an ordinal number of steps in general. Scott’s result is captured by the fact that in a CPO-enriched category, assuming that F is locally continuous, ω steps are sufficient for algebras given by projections.

Initial algebras and terminal coalgebras in many-sorted sets

We prove that the iterative construction of initial algebras converges for endofunctors F of many-sorted sets whenever F has an initial algebra. In the case of one-sorted sets, the convergence takes n steps where n is either an infinite regular cardinal or is at most 3. Dually, the existence of a many-sorted terminal coalgebra implies that the iterative construction of a terminal coalgebra converges. Moreover, every endofunctor with a fixed-point pair larger than the number of sorts is proved to have a terminal coalgebra. As demonstrated by James Worell, the number of steps here need not be a cardinal even in the case of a single sort: it is ω + ω for the finite power-set functor. The above results do not hold for related categories, such as graphs: we present non-constructive initial algebras and terminal coalgebras.

Coequational logic for accessible functors

Covarieties of coalgebras are those classes of coalgebras for an endofunctor H on the category of sets that are closed under coproducts, subcoalgebras and quotients. Equivalently, covarieties are classes of H-coalgebras that can be presented by coequations. Adámek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets. Here this result is extended to accessible functors: given a presentation of an accessible functor H, simple deduction systems for coequations are formulated and it is shown that regularity of the presentation implies soundness and completeness of these deduction systems. The converse is true whenever H has a non-trivial terminal coalgebra. Also a method is found to obtain concrete descriptions of cofree (and thus terminal) coalgebras of accessible functors, and is applied to the finite and countable powerset functor as well as to the finite distribution functor.

Approximating and computing behavioural distances in probabilistic transition systems

In an earlier paper we presented a pseudometric on the states of a probabilistic transition system, yielding a quantitative notion of behavioural equivalence. The behavioural pseudometric was defined via the terminal coalgebra of a functor based on a metric on Borel probability measures. In the present paper we give a polynomial-time algorithm, based on linear programming, to calculate the distances between states up to a prescribed degree of accuracy.

Terminal coalgebras and free iterative theories

Every finitary endofunctor H of Set can be represented via a finitary signature Σ and a collection of equations called “basic”. We describe a terminal coalgebra for H as the terminal Σ-coalgebra (of all Σ-trees) modulo the congruence of applying the basic equations potentially infinitely often. As an application we describe a free iterative theory on H (in the sense of Calvin Elgot) as the theory of all rational Σ-trees modulo the analogous congruence. This yields a number of new examples of iterative theories, e.g., the theory of all strongly extensional, rational, finitely branching trees, free on the finite power-set functor, or the theory of all binary, rational unordered trees, free on one commutative binary operation.

A behavioural pseudometric for probabilistic transition systems

Discrete notions of behavioural equivalence sit uneasily with semantic models featuring quantitative data, like probabilistic transition systems. In this paper, we present a pseudometric on a class of probabilistic transition systems yielding a quantitative notion of behavioural equivalence. The pseudometric is defined via the terminal coalgebra of a functor based on a metric on the space of Borel probability measures on a metric space. States of a probabilistic transition system have distance 0 if and only if they are probabilistic bisimilar. We also characterize our distance function in terms of a real-valued modal logic.

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.

De Bakker–Zucker processes revisited

The sets of compact and of closed subsets of a metric space endowed with the Hausdorff metric are studied. Both give rise to a functor on the category of 1-bounded metric spaces and nonexpansive functions. It is shown that the former functor has a terminal coalgebra and that the latter does not.

On a Description of Terminal Coalgebras and Iterative Theories

A concrete description of terminal coalgebras, T, of all finitary endofunctors of Set is presented: each such a functor is a quotient of the polynomial endofunctor of some finitary signature Σ modulo some “basic” equations. Then T can be described as the algebra of all infinite ∑-labeled trees modulo the congruence obtained by applying the basic equations finitely or infinitely many times (a concept defined below). As a consequence, free iterative theories in the sense of Calvin Elgot are described over all finitary endofunctors of Set: they are the theories of all rational ∑-labeled trees (i.e., trees having only finitely many subtrees) modulo the above congruence.

From modal logic to terminal coalgebras

We define a modal logic whose models are coalgebras of a polynomial functor. Bisimilarity turns out to be the same as logical equivalence. Ideas and concepts of modal logic are directly applied to the theory of coalgebras: we give an axiomatization and define canonical coalgebras. That leads to a completeness result. Each canonical coalgebra proves to be terminal in a certain class of coalgebras. The approach also yields a functional characterization of the terminal coalgebra of all coalgebras with respect to a given polynomial functor.

A Note on Hyperspaces and Terminal Coalgebras

The hyperspaces of closed and of compact subsets of an ultrametric space endowed with the Hausdorff metric are studied. Both give rise to a functor on the category of ultrametric spaces and nonexpansive functions. It is shown that the former functor does not have a terminal coalgebra and that the latter does.

Terminal sequences for accessible endofunctors

We consider the behaviour of the terminal sequence of an accessible endofunctor T on a locally presentable category K. The preservation of monics by T is sufficient to imply convergence, necessarily to a terminal coalgebra. We can say much more if K is Set, and κ is ω. In that case it is well known that we do not necessarily get convergence at ω, however we show that to ensure convergence we don't need to go to a higher cardinal, just to the next limit ordinal, ω + ω.For an ω-accessible endofunctor T on Set the construction of the terminal coalgebra can thus be seen as a two stage construction, with each stage being finitary. The first stage obtains the Cauchy completion of the initial T-algebra as the ω-th object in the terminal sequence Aω. In the second stage this object is pruned to get the final coalgebra Aω+ω. We give an example where Aω is the solution of the corresponding domain equation in the category of complete ultra-metric spaces.

Primitive (Co)Recursion and Course-of-Value (Co)Iteration, Categorically

In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic func- tion definition schemes of iteration and coiteration are modelled by constructions known as cata- morphisms and anamorphisms. Primitive recursion has been captured by a construction called paramorphisms. We draw attention to the dual construction of apomorphisms, and show on ex- amples that primitive corecursion is a useful function definition scheme. We also put forward and study two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of course-of-value iteration and its dual, respectively, and argue that even these are helpful.

Structural Induction and Coinduction in a Fibrational Setting

We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutionsX≅σ(X) whereXmay occur both positively and negatively in the type constructor σ. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunctions between “categories of algebras” (inserters).