Jiří Rosický

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.

Strongly complete logics for coalgebras

Coalgebras for a functor model different types of transition systems in a uni- form way. This paper focuses on a uniform account of finitary logics for set-based coalge- bras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts. Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor. Part II investigates algebras for a functor over ind-completions and extends the theorem of Jonsson and Tarski on canonical extensions of Boolean algebras with operators to this setting. Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T. Based on Part II we prove the logic to be strongly complete under a reasonable condition on T.

On quantales and spectra of C*-algebras

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max  is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

Left-determined model categories and universal homotopy theories

We call a model category left-determined if the weak equivalences are generated by the cofibrations. While simplicial sets are not left-determined, we show that their non-oriented variant is left-determined. This is used to give another and simpler proof of a recent result of D. Dugger about universal homotopy theories.

Notions of Lawvere theory

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products.

Cartesian closed exact completions

We characterize complete innitary extensive categories C having a cartesian closed exact completion. As an application, we show that the exact completion of the category of topological spaces is cartesian closed. It contributes to the study of equilogical spaces proposed by Scott (A new category? Domains, spaces and Equivalence Relations, preprint, 1996). c 1999 Elsevier Science B.V. All rights reserved. MSC: 18B99; 54A05

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

Characterizing spatial quantales

We will prove that an idempotent, right-sided quantale is spatial iff it is a subquantale of a product of simple quantales.

Weak factorizations, fractions and homotopies

Abstract We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

Concrete categories and infinitary languages

By definition, a concrete category is a category of sets which are endowed with an unspecified structure. There have been some attempts to make this structure specific. For example, Blanchard [6] used Bourbaki-type structures and Kurera and Pultr [17] have determined the structure by a functor Set-Set. Our aim is to consider concrete categories as categories of models of first-order theories. However, for these theories to be a syntactic counterpart of concrete categories, they must exceed the usual ones in the following three points: nonlogical symbols are of arbitrary arities, there may be a proper class of them and infinitary logical symbols are admitted. This language might be called ‘an unrestricted f.,,,‘. Its strength is illustrated by the fact that we may imagine any concrete category as a category consisting of models of this language. Of course, in this generality one cannot expect very model-theoretic results. Our approach is more an approach of ‘a working mathematician’, i.e. to see the relation between syntactic properties of theories and semantical properties of their categories of models. This approach has a practical aspect, i.e. to know what may be said about current models from life (e.g. when there exist limits, colimits, free objects or when models form a Cartesian closed category, a symmetric monoidal closed one etc.). There is also a theoretical aspect, i.e. to characterize theories providing a given categorical property or on the contrary to characterize concrete categories of models of theories of a prescribed kind. The pattern of these ‘categorical preservation theorems’ is the Beck-Linton theorem (see MacLane [20, p. 1471) which covers the equational case. A characteristic feature of these preservation theorems is the fact that they work outside any similarity type. The classical preservation theorem corresponding to the above mentioned Beck-Linton one is Birkhoff’s characteriz- ation of varieties of algebras of a given type. The generality of the syntax used means that the categories of models arising need not be legitimate. In Section 2 we establish a smallness property of a theory which ensures the legitimacy of its category of models. Moreover, these theories corres- pond to strongly fibre-small concrete categories in the sense of AdBmek, Herrlich 0022-4049/81/0000-OOOO/SO2.50 0 1981 North-Holland