Fundamental Constructions for Coalgebras, Corings, and Comodules

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.      

Universal categories of uniform and metric locales

Recall that a category is called universal if it contains an isomorphic copy of any concrete category as a full subcategory. In particular, if is universal then every monoid can be represented as the endomorphism monoid of an object in . A major obstacle to universality in categories of topological nature are the constant maps (which prevent, for instance, representing nontrivial groups as endomorphism monoids). Thus, to obtain, say, a universal category of uniform spaces, the constants have to be prohibited by artificial additional conditions (for instance, conditions of an openness type). Since in generalized spaces (locales) we do not necessarily have points, the question naturally arises as to whether we can get rid of surplus conditions in search of universality there. In this paper we prove that the category of uniform locales with all uniform morphisms is universal. Indeed we establish the universality already for the subcategory of very special uniform locales, namely Boolean metric ones. Moreover, universality is also obtained for more general morphisms, such as Cauchy morphisms, as well as for special metric choices of morphisms (contractive, Lipschitz). The question whether one can avoid uniformities remains in general open: we do not know whether the category of all locales with all localic morphisms is universal. However, the answer is final for the Boolean case: by a result of McKenzie and Monk ([10], see Section 4) one cannot represent groups by endomorphisms of Boolean algebras without restriction by an additional structure.We use only basic categorical terminology, say, that from the introductory chapters of [9]. All the necesasary facts concerning generalized spaces (frames, locales) and universality are explicitly stated. More detail on frames (locales) can be found in [8] and on universality and embeddings of categories in [11].Presented by E. Fried.     

Quotient morphisms, compositions, and Fredholm index

In this paper we define a composition between two quotient morphisms and prove a multiplication formula for the index of the composition of two Fredholm quotient morphisms. Using this formula and the fact that any linear relation can be seen as a quotient morphism, we obtain a multiplication formula for the index of the composition of two Fredholm linear relations.     

Topology of injective endomorphisms of real algebraic sets

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduce that any injective regular self-mapping of a real algebraic set is a homeomorphism. We show also a similar local property. All our results can be extended to arc-symmetric semi-algebraic sets and injective continuous arc-symmetric morphisms, and some results to Euler semi-algebraic sets and injective continuous semi-algebraic morphisms.Mathematics Subject Classification (2000):14Pxx, 14A10, 32B10     

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d−1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.     

Flag Higher Nash Blowups

In his previous article, the author has defined a higher version of the Nash blowup. In this article, we will introduce another higher version and prove that it is compatible with products and smooth morphisms. We will also prove that the product of curves can be desingularized via both versions.     

Linear systems, singularity and not too ramified morphisms

Given a smooth k-variety Y (where k is a field of arbitrary characteristic) and a linear systemL on Y we study the dimension of the singular locus of the general element ofL, both inside and outside the base locus B ofL. We interpret these results from the point of view of the transversality theory, and we improve a result by Speiser about the not too ramified morphisms. Moreover, we show that our results can be applied in some cases where a criterion by Zhang, for the smoothness of the general element ofL, fails. Entrata in Redazione il 15 settembre 1997, e in versione definitiva il 28 ottobre 1999.     

Nichtdifferenzierbare Morphismen differenzierbarer Räume

It is known ([4]) that under certain general conditions formal morphisms of differentiable spaces are differentiable. In this paper we show that these conditions are essentially necessary. Especially we establish the existence of formal, not differentiable morphisms of differentiable algebras. This solves a problem due to B. Malgrange ([1]).     

Bicompletion of metrically generated constructs

We develop a general technique to study completeness in a metrically generated construct X, the objects of which can be isomorphically described as sets endowed with a gauge of quasi metrics, saturated in a certain way. Our approach depends on a given functor F on X, describing the transition to some known topological construct A in which there already exists a completion theory described by some reflector R and for which also the associated firm class of morphisms $ \mathfrak{L} $ \mathfrak{L} (R) is known. This functor will permit to “lift” the notion of completeness from A to X. We describe sufficient conditions on R and F, ensuring that the lifting produces a completion theory R^F for the construct X₀ of T ₀ objects and that the associated firm class of morphisms can be derived from $ \mathfrak{L} $ \mathfrak{L} (R). Application of this technique to concrete examples results in many interesting completion theories and provides us with a far better understanding of several existing ones.     

关于态射的加权Moore-Penrose逆

研究了范畴中态射 f关于态射β和γ的加权 Moore-Penrose逆 fβ,γ+,分别给出了一般态射、有满单分解态射与有核 (上核 )的 fβ,γ+存在的充要条件及其相应的表达式 ,推广了 f关于对称态射β和γ的加权Moore-Penrose逆的相应结果 .     

Probabilization of Logics: Completeness and Decidability

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains.     

具有广义分解的态射的广义逆

本文给出了预加范畴中态射的广义分解的概念,并研究了具有广义分解的态射的｛１,ｉ｝－逆,Ｍｏｏｒｅ－Ｐｅｎｒｏｓｅ逆存在的条件及其表达式,得到了态射的群逆及Ｄｒａｚｉｎ逆存在的充要条件,推广了具有泛分解的态射的广义逆的相应结果．     

Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions.     

A characterization of the determinant

In this paper we examine group morphisms Λ: GL_n(R) → R^* from the general linear group over a commutative ring R into the group of units R^* of R and ask, "When are these morphisms functions of the determinant?"     

态射的加权Moore-Penrose逆

该文研究范畴中态射α关于对称态射β和γ的加权Ｍｏｏｒｅ Ｐｅｎｒｏｓｅ逆αβ,γ＋ ,分别给出了一般态射、有满单分解态射与有核（上核）态射的αβ,γ＋ 存在的充要条件及其相应的表达式．     

On Bicategories of Relations and Pullback Spans

In this paper, we study relations in general categories. Our approach requires that these must have finite products and factorization systems. Klein [5] has obtained a condition for composition of relations to be strictly associative. Here we consider the possibility that associativity only holds up to a coherent isomorphism, in other words that the relations are the morphisms of a bicategory in the sense of Bénabou [2].     

Modular Classes of Lie Algebroid Morphisms

We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.     

On Mono- and Epimorphisms in Varieties of Ordered Algebras

We study morphisms in varieties of ordered universal algebras. We prove that (i) monomorphisms are precisely the injective homomorphisms and that (ii) every regular monomorphism is an order embedding, but the converse is not true in general. We also give a necessary and sufficient condition for a morphism to be a regular epimorphism. Finally, we discuss factorizations in such varieties.     

Computing the partial word avoidability indices of ternary patterns

We study pattern avoidance in the context of partial words. The problem of classifying the avoidable binary patterns has been solved, so we move on to ternary and more general patterns. Our results, which are based on morphisms (iterated or not), determine all the ternary patternsʼ avoidability indices or at least give bounds for them.     

A General Method for Building Reflections

We establish a general method for generating reflections between categories. We then apply our technique to generate adjunctions starting from geometric morphisms between Grothendieck toposes; as particular cases, we recover various well-known Stone-type adjunctions and establish several new ones.