Bilimits are bifinal objects

We prove that a (lax) bilimit of a 2-functor is characterized by the existence of a limiting contraction in the 2-category of (lax) cones over the diagram. We also investigate the notion of bifinal object and prove that a (lax) bilimit is a limiting bifinal object in the 2-category of (lax) cones. Everything is developed in the context of marked 2-categories, so that the machinery can be applied to different levels of laxity, including pseudo-limits.     

On some constructions of algebraic objects

Mono-unary algebras may be used to construct homomorphisms, subalgebras, and direct products of algebras of an arbitrary type.     

On the notion of bimodel for functorial semantics

We discuss the notion of bimodel in order to obtain a classification of the equivalences between categories of models in the sense of functorial semantics.Research supported by NATO grant 900959 and FNRS grant 1.5.181.90F     

Order-adjoint monads and injective objects

Given a monad T on whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category, then the monad induced by the new situation is Kock-Zöberlein. Injective objects in the category of Kleisli monoids with respect to the class of initial morphisms then characterize the objects of the Eilenberg-Moore category of T, a fact that allows us to recuperate a number of known results, and present some new ones.     

Topological theories and closed objects

Recent work of several authors shows that many categories of interest to topologists can be represented as categories of lax algebras. In this paper we introduce the concept of a topological theory as a syntactical tool to deal with lax algebras, and show the usefulness of our approach by applying it to the study of function spaces.     

On an Essentially Algebraic Theory for Locally Presentable Categories

This paper presents a general construction, defining for each given strong generator in any locally finitely presentable category an essentially algebraic, finitary theory – maximal in a certain sense – such that is equivalent to the category of models of . For regular generators , generalization to the non-finitary case is easily done, and yields a new proof of the famous characterization of many-sorted quasivarieties.     

2-Cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors

In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in Elgueta (Adv. Math. 182 (2004) 204-277). We start by introducing a notion of 2-cosemisimplicial object in an arbitrary 2-category and analyzing the corresponding coherence question, where the permutohedra make their appearance. We then describe a general method to obtain usual cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in the 2-category of small K-linear categories and prove that the deformation complex introduced in Elgueta (to appear) can be obtained by this method from a 2-cosemisimplicial object that can be associated to . Finally, using this 2-cosemisimplicial object of and a generalization to the context of K-linear categories of the deviation calculus introduced by Markl and Stasheff for K-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to the integrability of an nth-order deformation of indeed correspond to cocycles in the third cohomology group , a question which remained open in Elgueta (Adv. Math. 182 (2004) 204-277).     

Lawvere theories enriched over a general base

We generalise the correspondence between Lawvere theories and finitary monads on in two ways. First, we allow our theories to be enriched in a category V that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable V-category A. We call the resulting notion that of a Lawvere A-theory. We extend the correspondence for ordinary Lawvere theories to one between Lawvere A-theories and finitary V-monads on A. We illustrate this with examples leading up to that of the Lawvere -theory for cartesian closed categories, i.e., the -enriched theory on the category for which the models are all small cartesian closed categories. We also briefly investigate change-of-base.     

Definable categories

We introduce the notion of a definable category–a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a 2-duality between the 2-category of small exact categories and the 2-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the 2-category of regular toposes and show that the diagram of 2-(anti-)equivalences between three 2-categories commutes; the corresponding additive triangle is well-known.     

On categorical notions of compact objects

Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.     

On the cobordism and commutative monoid with cancellation approaches to conformal field theory

In the late 1980s, Graeme Segal axiomatized conformal field theory in terms of a cobordism category. In that same preprint he outlined a more symmetric trace approach, which was recently rigorized in terms of pseudo algebras over a 2-theory. In this paper, we treat the cobordism approach in the pseudo algebra context. We introduce a new algebraic structure on a bicategory, called a pseudo 2-algebra over a theory, as a means of comparison for the two approaches. The main result states that the 2-category of pseudo algebras over a fixed 2-theory is biequivalent to the 2-category of pseudo 2-algebras over a fixed theory in certain situations.     

On the direct image of intersections in exact homological categories

Given a regular epimorphism f:X?Y in an exact homological category C, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

A useful fixpoint theorem

A sufficient condition for each extensive mapping of an ordered set to have a fixpoint is presented. Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 is gratefully acknowledged.     

On herstein's constructions relating jordan and associative superalgebras

We investigate the Jordan structure of a prime associative superalgebra and the Jordan structure of the symmetric elements of a *-prime associative superalgebra with superinvolution.     

Majorization and multiplier sequences

We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications.