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Jiří Adámek

Publications and source records attributed to Jiří Adámek.

At least 19 recordsLinked to original sources
arXiv2026

Terminal Coalgebras in Countably Many Steps

We present a collection of results that imply that an endofunctor on a category has a terminal coalgebra obtainable as a countable limit of its terminal-coalgebra chain. This holds for finitary endofunctors on locally finitely presentable categories under conditions on both the functor and the category. We adapt finiteness arguments that were originally advanced by Worrell concerning terminal coalgebras for finitary set functors. Examples include the categories of sets, posets, vector spaces, graphs, nominal sets, and presheaves on finite sets. Worrell also described, without proof, the terminal-coalgebra chain of the finite power-set functor. We provide a detailed proof following his ideas. We then turn to polynomial endofunctors on the categories of Hausdorff topological spaces and metric spaces. The Vietoris space of compact subsets yields an endofunctor $\mathscr{V}$ on the category of Hausdorff spaces. Vietoris polynomial endofunctors are built from $ \mathscr{V}$, the identity and constant functors by forming products, coproducts and compositions. Their terminal coalgebras are obtained in $ω$ steps. We then turn to the class of Hausdorff polynomial functors on the category of metric spaces, which is analogous but uses in lieu of $\mathscr{V}$ the Hausdorff functor $\mathcal{H}$. We prove they have terminal coalgebras obtained in $ω+ ω$ steps. Finally, we show that every finitary endofunctor on the category of vector spaces over a fixed field again has a terminal coalgebra obtained in $ω+ω$ steps.

cs.LO↗
arXiv2023

Sifted Colimits, Strongly Finitary Monads and Continuous Algebras

We characterize strongly finitary monads on categories $\mathsf{Pos}$, $\mathsf{CPO}$ and $\mathsf{DCPO}$ as precisely those preserving sifted colimits. Or, equivalently, enriched finitary monads preserving reflexive coinserters. We study sifted colimits in general enriched categories. For $\mathsf{CPO}$ and $\mathsf{DCPO}$ we characterize varieties of continuous algebras as precisely the monadic categories for strongly finitary monads.

math.CT↗
arXiv2023

Varieties of ordered algebras as categories

A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere's classical characterization of varieties of ordinary algebras. We also prove that varieties of ordered algebras are precisely the free completions of duals of discrete Lawvere theories under sifted colimits.

math.CT↗
arXiv2023

Varieties of Quantitative or Continuous Algebras (Extended Abstract)

Quantitative algebras are algebras enriched in the category $\mathsf{Met}$ of metric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka $1$-basic varieties) as classes of quantitative algebras presented by quantitative equations. We prove that they bijectively correspond to strongly finitary monads $T$ on $\mathsf{Met}$. This means that $T$ is the Kan extension of its restriction to finite discrete spaces. An analogous result holds in the category $\mathsf{CMet}$ of complete metric spaces. Analogously, continuous algebras are algebras enriched in $\mathsf{CPO}$, the category of $ω$-cpos, so that all operations are continuous. We introduce equations between extended terms, and prove that varieties (classes presented by such equations) correspond bijectively to strongly finitary monads $T$ on $\mathsf{CPO}$. This means that $T$ is the Kan extension of its restriction to finite discrete cpos. (The two results have substantially different proofs.) An analogous result is also presented for monads on $\mathsf{DCPO}$. We also characterize strong finitarity in all the categories above by preservations of certain weighted colimits. As a byproduct we prove that directed colimits commute with finite products in all cartesian closed categories.

math.CT↗
arXiv2022

An Initial Algebra Theorem Without Iteration

The Initial Algebra Theorem by Trnková et al.~states, under mild assumptions, that an endofunctor has an initial algebra provided it has a pre-fixed point. The proof crucially depends on transfinitely iterating the functor and in fact shows that, equivalently, the (transfinite) initial-algebra chain stops. We give a constructive proof of the Initial Algebra Theorem that avoids transfinite iteration of the functor. For a given pre-fixed point $A$ of the functor, it uses Pataraia's theorem to obtain the least fixed point of a monotone function on the partial order formed by all subobjects of $A$. Thanks to properties of recursive coalgebras, this least fixed point yields an initial algebra. We obtain new results on fixed points and initial algebras in categories enriched over directed-complete partial orders, again without iteration. Using transfinite iteration we equivalently obtain convergence of the initial-algebra chain as an equivalent condition, overall yielding a streamlined version of the original proof.

cs.LO↗
arXiv2021

Approximate injectivity and smallness in metric-enriched categories

Properties of categories enriched over the category of metric spaces are investigated and applied to a study of constructions known from that category and the category of Banach spaces. For every class of morphisms satisfying a mild smallness condition we prove the corresponding approximate-injectivity class is weakly reflective, and we study the properties of the reflection morphisms. As an application we present a new categorical proof of the essential uniqueness of the Gurarii space.

math.CT↗
arXiv2021

On the Behaviour of Coalgebras with Side Effects and Algebras with Effectful Iteration

For every finitary monad $T$ on sets and every endofunctor $F$ on the category of $T$-algebras we introduce the concept of an ffg-Elgot algebra for $F$, that is, an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad $T$. The goal is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point $φF$, i.e. the colimit of all $F$-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of $T$-algebras.

cs.LO↗
arXiv2021

Algebraic cocompleteness and finitary functors

A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal CPO-completion. And they also both carry a canonical ultrametric with the same Cauchy completion.

math.CT↗
arXiv2020

Finitary Monads on the Category of Posets

Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on $\mathsf{Pos}$ are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.

math.CT↗
arXiv2020

On Well-Founded and Recursive Coalgebras

This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor's General Recursion Theorem that every well-founded coalgebra is recursive, and we study under which hypothesis the converse holds. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

cs.LO↗
arXiv2019

On Finitary Functors

A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is equivalent to $F$ being finitary for all functors between `reasonable' locally finitely presentable categories, provided that $F$ preserves monomorphisms. We also discuss the question when that last assumption can be dropped. The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any finite group either on Set or on K-Vec for fields K of characteristic 0. All this generalizes to locally $λ$-presentable categories, $λ$-accessible functors and $λ$-presentable algebras. As an application we obtain an easy proof that the Hausdorff functor on the category of complete metric spaces is $\aleph_1$-accessible.

math.CT↗
arXiv2019

Finitely Presentable Algebras For Finitary Monads

For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by finitely many generators and finitely many relations.

math.CT↗
arXiv2019

How nice are free completions of categories?

Every category $\mathcal K$ has a free completion $\mathcal P \mathcal K$ under colimits and a free completion $Σ\mathcal K$ under coproducts. A number of properties of $\mathcal K$ transfer to $\mathcal P \mathcal K$ and $Σ\mathcal K$ (e.g., completeness or cartesian closedness). We prove that $\mathcal P\mathcal K$ is always a pretopos, but, for $\mathcal K$ large, seldom a topos. Moreover, for complete categories $\mathcal K$ we prove that $\mathcal P\mathcal K$ is locally cartesian closed whenever $\mathcal K$ is additive or cartesian closed or dual to an extensive category. We also study the question whether $\mathcal P \mathcal K$ is (co)wellpowered. The answer is affirmative for "set-like" categories. But for a number of categories $\mathcal K$ the answer turns out to be negative.

math.CT↗
arXiv2017

On Corecursive Algebras for Functors Preserving Coproducts

For an endofunctor $H$ on a hyper-extensive category preserving countable coproducts we describe the free corecursive algebra on $Y$ as the coproduct of the final coalgebra for $H$ and the free $H$-algebra on $Y$. As a consequence, we derive that $H$ is a cia functor, i.e., its corecursive algebras are precisely the cias (completely iterative algebras). Also all functors $H(-) + Y$ are then cia functors. For finitary set functors we prove that, conversely, if $H$ is a cia functor, then it has the form $H = W \times (-) + Y$ for some sets $W$ and $Y$.

cs.LO↗
arXiv2017

Eilenberg Theorems for Free

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Bojańczyk and of Adámek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.

cs.FL↗
arXiv2014

Coproducts of Monads on Set

Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra. We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective. Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.

cs.LO↗
arXiv2014

Colimits of Monads

The category of all monads over many-sorted sets (and over other "set-like" categories) is proved to have coequalizers and strong cointersections. And a general diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In contrast, coequalizers fail to exist e.g. for monads over the (presheaf) category of graphs. For more general categories we extend the results on coproducts of monads from [2]. We call a monad separated if, when restricted to monomorphisms, its unit has a complement. We prove that every collection of separated monads with arbitrarily large joint pre-fixpoints has a coproduct. And a concrete formula for these coproducts is presented.

cs.LO↗