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Abstract We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated coherence conditions arising from polynomial composition; however, in this paper we avoid most of these coherence conditions using the properties of generic bicategories. In addition, we give a new proof of the universal properties of the bicategory of spans, and also establish the universal properties of the bicategory of spans with invertible 2-cells; showing how these properties may be used to describe the universal properties of polynomials. [ABSTRACT FROM AUTHOR]

We fix any bicategory A together with a class of morphisms W A , such that there is a bicategory of fractions A [ W A − 1 ] (as described by D. Pronk). Given another such pair ( B , W B ) and any pseudofunctor F : A → B , we find necessary and sufficient conditions in order to have an induced pseudofunctor G : A [ W A − 1 ] → B [ W B − 1 ] . Moreover, we give a simple description of G in the case when the class W B is “right saturated”. [ABSTRACT FROM AUTHOR]

Algebraic weak factorisation systems ( awfs ) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad–monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs —drawing on work of previous authors—and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs , and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation. [ABSTRACT FROM AUTHOR]

In this paper, we provide some formulae to calculate orbital characters. Especially we give a lifting theorem by which one can read off the value of orbital characters from local subgroups. With orbital characters, we can show that Reynolds ideal is dual to the vector space of simple modules and this leads to a new method to count simple modules. We also give a natural morphism from local to general vector space of simple modules and determine it completely. Some other applications to modular representation are also included. [ABSTRACT FROM AUTHOR]

Abstract: An algebraically exact category is one that admits all of the limits and colimits which every variety of algebras possesses and every forgetful functor between varieties preserves, and which verifies the same interactions between these limits and colimits as hold in any variety. Such categories were studied by Adámek, Lawvere and Rosický: they characterised them as the categories with small limits and sifted colimits for which the functor taking sifted colimits is continuous. They conjectured that a complete and sifted-cocomplete category should be algebraically exact just when it is Barr-exact, finite limits commute with filtered colimits, regular epimorphisms are stable by small products, and filtered colimits distribute over small products. We prove this conjecture. [Copyright &y& Elsevier]

Abstract: It is shown that (1) the property of a category of having unique essential completions is inherited by any monocoreflective subcategory whose monics are monic in the larger category provided the essential monomorphisms of the latter satisfy a certain condition, and (2) the category of archimedean -groups with weak order unit is of that type. Then, identification is made of various such subcategories of and their essential completions are analysed. The methods exploit the functors from these categories to categories of frames. [Copyright &y& Elsevier]

Abstract: By a multi-pointed category we mean a category equipped with an ideal of null morphisms, i.e. a class of morphisms satisfying for any composable pair of morphisms. Such categories are precisely the categories enriched in the category of pairs where is a set and is a subset of , whereas a pointed category has the same enrichment, but restricted to those pairs where is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories. [Copyright &y& Elsevier]

Abstract: The nerve of a strict omega-category is a simplicial set with additional structure, making it into a so-called complicial set, and strict omega-categories are in fact equivalent to complicial sets. The nerve functor is represented by a sequence of strict omega-categories, called orientals, which are associated to simplexes. In this paper, we give a detailed algebraic description of the morphisms between orientals. The aim is to describe complicial sets algebraically, by operators and equational axioms. [Copyright &y& Elsevier]

Abstract: We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg–Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. We explain at the end of the paper that a similar phenomenon occurs in many other situations. [Copyright &y& Elsevier]

Abstract: Let be a regular category with pushouts of regular epimorphisms by regular epimorphism and the category of regular epimorphisms in . We prove that every regular epimorphism in is an effective descent morphism if, and only if, is a regular category. Then, moreover, every regular epimorphism in is an effective descent morphism. This is the case, for instance, when is either exact Goursat, or ideal determined, or is a category of topological Mal’tsev algebras, or is the category of -fold regular epimorphisms in any of the three previous cases, for any . [Copyright &y& Elsevier]

Abstract: The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve assertions of the existence and uniqueness of certain arrows. Weak notions arise when one drops the uniqueness requirement and asks only for existence. The enriched versions of the usual notions involve certain morphisms between hom-objects being invertible; here we introduce enriched versions of the weak notions by asking that the morphisms between hom-objects belong to a chosen class of “surjections”. We study in particular injectivity (weak orthogonality) in the enriched context, and illustrate how it can be used to describe homotopy coherent structures. [Copyright &y& Elsevier]

Abstract: Length preserving morphisms and inverse of substitutions are two well-studied operations on regular languages. Their connection with varieties generated by power monoids was established independently by Reutenauer and Straubing in 1979. More recently, an ordered version of this theory was proposed by Polák and by the authors. In this paper, we present an improved version of these results and obtain the following consequences. Given a variety of finite ordered monoids , let be the variety of finite ordered monoids generated by the upper set monoids of members of . Then . This contrasts with the known results for the unordered case: the operator corresponding to power monoids satisfies , but the varieties , , and can be distinct. [Copyright &y& Elsevier]

Abstract: Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale satisfying mild algebraic properties, the sheaves on can be defined to be the idempotent self-adjoint -valued matrices. These can be thought of as -valued equivalence relations, and, accordingly, the morphisms of sheaves are the -valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic étale groupoid (the classifying topos of ) is equivalent to the category of sheaves on its involutive quantale . As a means toward this end, we begin by replacing the category of matrix sheaves on by an equivalent category of complete Hilbert -modules, and we approach the envisaged example where is an inverse quantal frame by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module-theoretic description of arbitrary actions of étale groupoids, both of which may be interesting in their own right. [Copyright &y& Elsevier]

Abstract: We study necessary and sufficient conditions for the existence of irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the -power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary -category with tilting object, such irreducible morphisms exist if and only if is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms. [Copyright &y& Elsevier]

Abstract: We give a characterization of those finitely complete categories with initial object and pushouts of split monomorphisms that admit categorical semidirect products. As an application we examine the case of groupoids with fixed set of objects. Further, we extend this to the internal case. [Copyright &y& Elsevier]

Abstract: It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle on defined as the kernel of a general epimorphism is (semi)stable. In this note we restrict our attention to the case of syzygy bundles on associated to generic forms of the same degree . Our first goal is to prove that is stable if and . This bound improves, in general, the bound given by Hein (2008 ), Appendix A. Display Omitted In the last part of the paper, we study moduli spaces of stable rank vector bundles on containing syzygy bundles. We prove that if , and , then the syzygy bundle is unobstructed and it belongs to a generically smooth irreducible component of dimension , if , and , if . [Copyright &y& Elsevier]

Abstract: We study the left degree of an irreducible morphism with and indecomposable modules in a standard component of the Auslander-Reiten quiver, for . Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two. [Copyright &y& Elsevier]

Abstract: This paper focuses on a property of enriched functors reflecting the factorisation of morphisms, used in concurrency semantics. According to Lawvere [F.W. Lawvere, State categories and response functors, 1986, Unpublished manuscript], a functor strictly reflecting morphism factorisation induces a notion of state on its domain, when it is considered as a control functor. This intuition works both in case of physical and computing processes [M. Bunge, M.P. Fiore, Unique factorisation lifting functors and categories of linearly-controlled processes, Math. Structures Comput. Sci. 10 (2) 2000 137–163; M.P. Fiore, Fibered models of processes: Discrete, continuous and hybrid systems, in: Proc. of IFIP TCS 2000, in: LNCS, vol. 1872, 2000, pp. 457–473]. In this note we investigate a more general property in the family of models we proposed elsewhere for communicating processes, and we assess their bisimulation relations [S. Kasangian, A. Labella, Observational trees as models for concurrency, Math. Structures. Comput. Sci. 9 (1999) 687–718; R. De Nicola, D. Gorla, A. Labella, Tree-Functors, determinacy and bisimulations, Technical Report, 02/2006, Dip. di Informatica, Univ. di Roma “La Sapienza” (Italy), 2008 (submitted for publication), http://www.dsi.uniroma1.it/%7Egorla/papers/DGL-TR0206.pdf]. Hence, we adapt the notion of “Conduché condition” [F. Conduché, Au sujet de l’existence d’adjoints à droîte aux foncteurs image reciproque dans la catégorie des catégories, C. R. Acad. Sci. Paris 275 (1972) A891–894] to the context of enriched category theory. This notion, weaker than the original “Moebius condition” used by Lawvere, seems to be more suitable for the description of the concurrency models parametrised w.r.t. a base category via the mechanism of change of base, actually. The base category is a monoidal 2-category; a category of generalised trees, , is obtained from it. We consider Conduché -based categories, where enrichment reflects factorisation of objects in the base category. We prove that a form of Conduché’s theorem holds for Conduché -functors. We also show how the Conduché condition plays a crucial role in modelling concurrent processes and bisimulations between them. The notions of “state preservation” and “determinacy” [R. Milner, Communication and Concurrency, Prentice Hall International, 1989] are formally characterised. [Copyright &y& Elsevier]

Abstract: We construct a monomorphism from the 2-compact group into a 2-compact unitary group. [Copyright &y& Elsevier]

Abstract: Stochastic relations are the Kleisli morphisms for the Giry monad. This paper proposes the study of the associated morphisms and congruences. The relationship between kernels of these morphisms and congruences is studied, and a unique factorization of a morphism through this kernel is shown to exist. This study is based on an investigation into countably generated equivalence relations on the space of all subprobabilities. Operations on these relations are investigated quite closely. This utilizes positive convex structures and indicates cross-connections to Eilenberg–Moore algebras for the Giry monad. Hennessy–Milner logic serves as an illustration for randomized morphisms and congruences. [Copyright &y& Elsevier]

Abstract: -structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is the extension of a -algebra by an associative algebra. Actions of -algebras on associative and -algebras are analyzed; extensions of -algebras by associative and -algebras that they act upon are constructed. The resulting -algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the -algebra on the deformation complex of the corresponding diagram algebra. [Copyright &y& Elsevier]

Abstract: Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591–3622], we show that a Frobenius morphism on an algebra induces naturally a functor on the (bounded) derived category of , and we further prove that the derived category of for the -fixed point algebra is naturally embedded as the triangulated subcategory of -stable objects in . When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander–Reiten triangles in and those in . Thus, the AR-quiver of can be obtained by folding the AR-quiver of . Finally, we further extend this relation to the root categories of and of , and show that, when is hereditary, this folding relation over the indecomposable objects in and results in the same relation on the associated root systems as induced from the graph folding relation. [Copyright &y& Elsevier]