Quantitative algebras are $Σ$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces such varieties bijectively correspond to strongly finitary monads on $\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric spaces, provided that strongly finitary endofunctors are closed under composition. For uncountable cardinals $λ$ there is an analogous bijection between varieties of $λ$-ary quantitative algebras and monads that are strongly $λ$-accessible. Moreover, we present a bijective correspondence between $λ$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $λ$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $λ$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented.

Abstract It is well known that classical varieties of $\Sigma$ -algebras correspond bijectively to finitary monads on $\mathsf{Set}$ . We present an analogous result for varieties of ordered $\Sigma$ -algebras, that is, categories of algebras presented by inequations between $\Sigma$ -terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$ . That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\mathsf{Set}$ . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on $\mathsf{Set}$ to strongly finitary monads on $\mathsf{Pos}$ .

For a commutative quantale $\mathcal{V}$, the category $\mathcal{V}-cat$ can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor $T$ (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor $T_{\mathcal{V}}$ on $\mathcal{V}-cat$. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation $X\cong TX$ in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base. Mathematically, the heart of the matter is to show that, for any commutative quantale $\mathcal{V}$, the `discrete' functor $D:\mathsf{Set}\to \mathcal{V}-cat$ from sets to categories enriched over $\mathcal{V}$ is $\mathcal{V}-cat$-dense and has a density presentation that allows us to compute left-Kan extensions along $D$. Comment: 57 pages; extended version of the paper presented at CALCO 2015; accepted for publication in LMCS; Sections 2.4 and 3.3 were added

Positive modal logic, as introduced by Dunn in 1995, is the negation-free fragment of the standard modal logic of all Kripke frames. Positive coalgebraic logic, introduced by the authors in a previous work, expands the above result from Kripke frames to more general transition systems, namely to coalgebras of weak-pullback preserving functors. We show that this construction is both modular and uniform in the functor giving the type of coalgebra. More precisely, we formalize both Set and Pos-based coalgebraic modal logic as institutions, and we exhibit a morphism of institutions between them giving the positive fragment of coalgebraic modal logic.

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).

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.

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1. For every class $\mathcal{H}$ of morphisms, we study the subcategory of all objects that are Kan-injective with respect to $\mathcal{H}$ and all morphisms preserving Kan extensions. For categories such asTop0andPos, we prove that whenever $\mathcal{H}$ is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings inTop0for which Kan-injectivity does not yield a monadic category.

Bloom and Ésik's concept of iteration theory summarises all equational properties that iteration has in common applications, for example, in domain theory, where to every system of recursive equations, the least solution is assigned. This paper shows that in the coalgebraic approach to iteration, the more appropriate concept is that of a functorial iteration theory (called Elgot theory). These theories have a particularly simple axiomatisation, and all well-known examples of iteration theories are functorial. Elgot theories are proved to be monadic over the category of sets in context (or, more generally, the category of finitary endofunctors of a locally finitely presentable category). This demonstrates that functoriality is an equational property from the perspective of sets in context. In contrast, Bloom and Ésik worked in the base category of signatures rather than sets in context, and there iteration theories are monadic but Elgot theories are not. This explains why functoriality was not included in the definition of iteration theories.

We fix a logical connection (Stone ˧ Pred : Setop → BA given by 2 as a schizophrenic object) and study coalgebraic modal logic that is induced by a functor T: Set → Set that is finitary and standard and preserves weak pullbacks and finite sets. We prove that for any such T, the cover modality nabla is a left (and its dual delta is a right) adjoint relative to ω. We then consider monotone unary modalities arising from the logical connection and show that they all are left (or right) adjoints relative to ω.

We propose a construction of the final coalgebra for a finitary endofunctor of a finitely accessible category and study conditions under which this construction is available. Our conditions always apply when the accessible category is cocomplete, and is thus a locally finitely presentable (l.f.p.) category, and we give an explicit and uniform construction of the final coalgebra in this case. On the other hand, our results also apply to some interesting examples of final coalgebras beyond the realm of l.f.p. categories. In particular, we construct the final coalgebra for every finitary endofunctor on the category of linear orders, and analyse Freyd's coalgebraic characterisation of the closed unit as an instance of this construction. We use and extend results of Tom Leinster, developed for his study of self-similar objects in topology, relying heavily on his formalism of modules (corresponding to endofunctors) and complexes for a module.

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.

Iterative monads were introduced by Calvin Elgot in the 1970's 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.

Denotational semantics can be based on algebras with additional structure (order, metric, etc.) which makes it possible to interpret recursive specifications. It was the idea of Elgot to base denotational semantics on iterative theories instead, i.e., theories in which abstract recursive specifications are required to have unique solutions. Later Bloom and Esik studied iteration theories and iteration algebras in which a specified solution has to obey certain axioms. We propose so-called Elgot algebras as a convenient structure for semantics in the present paper. An Elgot algebra is an algebra with a specified solution for every system of flat recursive equations. That specification satisfies two simple and well motivated axioms: functoriality (stating that solutions are stable under renaming of recursion variables) and compositionality (stating how to perform simultaneous recursion). These two axioms stem canonically from Elgot's iterative theories: We prove that the category of Elgot algebras is the Eilenberg-Moore category of the monad given by a free iterative theory.

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

By the Final Coalgebra Theorem of Aczel and Mendler, every endofunctor of the category of sets has a final coalgebra, which, however, may be a proper class. We generalise this to all ‘well-behaved’ categories .

Every finitary endofunctor of $\Set$ is proved to generate a free iterative theory in the sense of Elgot. This work is based on coalgebras, specifically on parametric corecursion, and the proof is presented for categories more general than just $\Set$ .