Diads and their Application to Topoi

It is well known that the category of coalgebras for a finite-limit preserving comonad on a topos is again a topos, and the category of algebras for a finite-limit preserving monad is a topos if the monad is idempotent, but not in general. A generalisation of this result (Paré et al., Bull Aus Math Soc 39(3):421–431, 1989) is that the full subcategory of fixed points for any idempotent finite-limit preserving endofunctor is again a topos (and indeed a subquotient in the category of topoi and geometric morphisms). Here, we present a common generalisation of all the above results, based on a notion which we call a diad, which is a common generalisation of a monad and a comonad. Many of the constructions that can be applied to monads and comonads can be extended to all diads. In particular, the category of algebras or coalgebras can be generalised to a category of dialgebras for a diad. The generalisation we present here is that the category of dialgebras for a finite-limit preserving left diad (for example, the diad corresponding to a comonad, or any idempotent endofunctor) on a topos is again a topos.     

A note on triangulated monads and categories of module spectra

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg–Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is ‘essentially monadic’, i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is ‘intrinsically monadic’. It follows that for any highly structured ring spectrum, its category of homotopy (aka naïve) modules is triangulated if and only if it is equivalent to its category of highly structured (aka strict) modules.     

拓扑空间范畴、拓扑FUZZ范畴与拓扑分子格范畴间的反射与余反射

从整体角度出发，证明了拓扑空间范畴Ｔｏｐ分别是拓扑Ｆｕｚｚ范畴ＴｏｐＦｕｚ与拓扑分子格范畴ＴＭＬ的反射与余反射满子范畴，ＴｏｐＦｕｚ是ＴＭＬ的反射与余反射（非满）子范畴．     

On the boundedness of topological categories

Let K be a complete and cocomplete category with a given proper (E,M)-factorization. K is called well-bounded if K is moreover bounded with a generator and cowellpowered with respect to the given factorization. Freyd-Kelly proved the following theorem about well-bounded categories: Let K be a well-bounded category and let Γ be a class of cylinders in the small category C¹, and let all but a set of these cylinders be cones. Then Γ(C,K) is a reflective subcategory of [C,K]. The main results of this paper are: (I) If F: K→L is a Top-functor and L is well-bounded, then K is well-bounded. (II) If U is an E-reflective subcategory of a well-bounded category,then U is again wellbounded. As a corollary one obtains for instance that all coreflective and all epireflective subcategories of the category of topological spaces are well-bounded.     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.     

L-预拓扑空间中的近似L-开集及相关的连通性和范畴性质

研究L-预拓扑空间中L-开集的一些近似形式(其中L是有逆合对应的C-格):θ-L-开集,δ-L-开集,α-L-开集以及β-L-开集。证明了(1)当L的最大元1是并既约元时L-预拓扑空间的连通性、θ-连通性、δ-连通性和α-连通性是等价的;(2)θLTop是LTopsθ、LTop以及LTopθ的反射满子范畴,δLTop是LTopsδ和LTopδ的反射满子范畴;(3)当格L为幂集格时αLTop是LTopsα和LTopα的余反射满子范畴;θLPTop是LPTopsθ、LPTop以及LPTopθ的反射满子范畴,δLPTop是LPTopsδ和LPTopδ的反射满子范畴。     

Note on the orderability of idempotent semigroups

In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saitô gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.     

A NOTE ON SUBCATEGORIES OF REACHABLE MACHINES

Abstract

Given a category K. a state-behaviour process X: K → K and an image factorization system (E,M) for K, it is shown that under mild conditions the full subcategory with objects all reachable machines is coreflective. A corresponding result is obtained for coequalizer-reachable machines if X is only an input process.     

On a Problem of M. Kambites Regarding Abundant Semigroups

A semigroup is regular if it contains at least one idempotent in each ?-class and in each ?-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ?-class and in each ?-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ?*-class and in each ?*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ?* and ?*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ?* and ?*-class, must the idempotents commute? In this note, we provide a negative answer to this question.     

TOPOLOGY WITHOUT POINTS

Abstract

Other theories that develop topology without points are either excessively artificial or suffer from a lack of rigor. In this paper it is assumed that worlds W are composed of parts that form a complete Boolean algebra [xbar] and that the collection [Wbar] of all points of W is a certain subcollection of all filters defined over [xbar]. Two axioms are given for points which, given suitable definitions, convert [Wbar] into a compact Hausdorff space. Nearness collections of parts of W are defined which satisfy all the axioms of Herrlich for nearness except that closure is defined without mentioning points and consequently one may define closed and open parts. A category of worlds is defined in which the objects are lattices of closed parts of a world and the arrows are roughly speaking the far-preserving mps. It is shown that the category of compact T₁-spaces is a reflective subcategory of the category of worlds.     

Heredity and Coreflective Subcategories of the Category of Topological Spaces

Every hereditary coreflective subcategory of Top containing the category of finitely-generated spaces is shown to be generated by a class of spaces having a unique accumulation point. It is also shown that the coreflective hull of a union of two hereditary coreflective subcategories of Top need not be hereditary so that a coreflective subcategory of Top need not have a hereditary coreflective kernel.     

ON THE ADJOINT SITUATIONS BETWEEN BEHAVIOUR AND REALIZATION

ABSTRACT

Following the Arbib-Manes approach to machines in categories, the categories Mach(C) of machines and TBeh(C) of total behaviours are constructed, where c is a class of state-behaviour processes. It is shown that the total external behaviour functor E: Mach(C) → TBeh(C) has a left adjoint and that the free realization of any total behaviour is reachable. Furthermore, the restriction of E to the full subcategory of Mach(C) with all reachable machines as objects has as right adjoint the minimal realization functor.     

Idempotent probability measures on ultrametric spaces

Following the construction due to Hartog and Vink we introduce a metric on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. We prove that this functor is the functorial part of a monad on this category. This monad turns out to contain the hyperspace monad.     

The decomposition of idempotents associated with inflators

In a recent paper, Friedland, Hershkowitz, and Schneider introduced a new matrix product called the inflation product and a new class of matrices called inflators. Fundamental to the constructions were certain idempotent matrices associated with the inflators. This paper studies the structure of the idempotent matrix associated with an inflator. In particular, it is shown that if the idempotent associated with an inflator has rank greater than one, then the idempotent can be split into several pairwise orthogonal idempotents of lower rank such that the resultant idempotents are associated with inflators which are inflation product factors of the original inflator. The indecomposable idempotents associated with the decomposition of an inflator are characterized in terms of rank, and are shown to be generally nonunique. The number of indecomposable idempotents in a splitting is shown to be invariant.     

PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

On exchange rings with primitive factor rings artinian

Abstract

In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices.     

Representation theorems for directed completions of consistent algebraic L-domains

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain up to isomorphism. It is also proved that the category CALDOM of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category ALDOM of algebraic L-domains and Scott continuous maps as a full reflective subcategory. Received January 8, 2005; accepted in final form June 15, 2005.     

Z-Quantale及其范畴性质

本文把集系统的概念应用到Quantale理论中,作为Quantale的一般化,引入了Z-quantale的概念,研究了Z-quantale及其范畴的若干性质.主要结果有:证明了Z-quantale范畴是序半群范畴的反射子范畴,凝聚Z-quantale范畴是Z-quantale范畴的余反射子范畴.讨论了Z-quantale范畴中的投射对象,证明了Z-quantale A是E-投射的当且仅当它是稳定Z-连续的.