Jiří Adámek

On the greatest fixed point of a set functor

Abstract The greatest fixed point of a set functor is proved to be (a) a metric completion and (b) a CPO-completion of finite iterations. For each (possibly infinitary) signature Σ the terminal Σ-coalgebra is thus proved to be the coalgebra of all Σ-labelled trees; this is the completion of the set of all such trees of finite depth. A set functor is presented which has a fixed point but does not have a greatest fixed point. A sufficient condition for the existence of a greatest fixed point is proved: the existence of two fixed points of successor cardinalities.

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.

Least Fixed Point of a Functor

We exhibit a con- struction of the LFP, generalizing the Knaster-Tarski formula lub(F”(O)},,, : lubs are substituted by well-ordered colimits and n is allowed to be an arbitrary ordinal. Related LFP constructions, always restricted to 1z E w, have been considered by various authors [6, 11, 13, 141. The advantage of the present approach is its effectiveness: Whenever a functor

From Varieties of Algebras to Covarieties of Coalgebras

Abstract Varieties of F-algebras with respect to an endofunctor F on an arbitrary cocomplete category C are defined as equational classes admitting free algebras. They are shown to correspond precisely to the monadic categories over C . Under suitable assumptions satisfied in particular by any endofunctor on Set and Set op the Birkhoff Variety Theorem holds. By dualization, covarieties over complete categories C are introduced, which then correspond to the comonadic categories over C , and allow for a characterization in dual terms of the Birkhoff Variety Theorem. Moreover, the well known conditions of accessibilitly and boundedness for Set -functors F, sufficient for the existence of cofree F-coalgebras, are shown to be equivalent.

A coalgebraic perspective on minimization and determinization

Coalgebra offers a unified theory of state based systems, including infinite streams, labelled transition systems and deterministic automata. In this paper, we use the coalgebraic view on systems to derive, in a uniform way, abstract procedures for checking behavioural equivalence in coalgebras, which perform (a combination of) minimization and determinization. First, we show that for coalgebras in categories equipped with factorization structures, there exists an abstract procedure for equivalence checking. Then, we consider coalgebras in categories without suitable factorization structures: under certain conditions, it is possible to apply the above procedure after transforming coalgebras with reflections. This transformation can be thought of as some kind of determinization. We will apply our theory to the following examples: conditional transition systems and (non-deterministic) automata.

Presentation of Set Functors

Accessible set functors can be presented by signatures and equations as quotients of polynomial functors.We determine how preservation of pullbacks and other related properties (often applied in coalgebra) are reflected in the structure of the system of equations.

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.

Recursive coalgebras of finitary functors

For finitary set functors preserving inverse images, recursive coalgebras A of Paul Taylor are proved to be precisely those for which the system described by A always halts in finitely many steps.

On injectivity in locally presentable categories

AbstractWe show that some fundamental results about projectivity classes, weakly coreflective subcate-gories and cotorsion theories can be generalized from R -modules to arbitrary locally presentablecategories.  2004 Elsevier Inc. All rights reserved. 1. IntroductionInjectivity in locally presentable categories is well understood(see [2]). The basic resultis that a full subcategory A of a locally presentable category K is a small-injectivity class(i.e., there is a set M of morphisms of K such that A consists of all objects injective w.r.t.each morphism in M ) if and only if A is accessible and closed in K under products and λ -directed colimits for some regular cardinal λ . Accessibility of A can be replaced by A beingalsoclosed under λ -puresubobjects.Here, λ -puresubobjectsare precisely λ -directedcolimits of split subobjects. This result was re-proved for additive locally presentablecategories by H. Krause [13]. Injectivity classes are closely related to weakly reflectivesubcategories. Every small-injectivity class of a locally presentable category