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1.
《Quaestiones Mathematicae》2013,36(1-3):159-175
Abstract If a functor U has a left co-unadjoint then U can be factored through a category of semad algebras. An analogue of the Beck monadicity theory is obtained. If R is a ring without a left unit but satisfying R2 = R then the category of unitary left R-modules need not be monadic over Set. The forgetful functor has, however, a left co-unadjoint for which a comparison functor is an equivalence of categories. Another example of a semadic functor is obtained by composing the forgetful functor from Abelian groups to Set with the doubling functor. The semi-adjoint situations in the senses of Medvedev and Davis are examined. 相似文献
2.
《Quaestiones Mathematicae》2013,36(1-3):59-84
This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other. 相似文献
3.
《Quaestiones Mathematicae》2013,36(4):303-312
Abstract This paper deals with projectives (in the sense of K.A.Hardie [5] relative to a right adjoint functor U: A → K. We answer the question, raised by R.-E. Hoffmann [6] p. 135, of knowing under what conditions there exists an equivalence between Proj u and Proj Ur, induced by the comparison functor Φ: A → KT, where T denotes the monad induced by U. In the case, that U is an algebraic functor we also give necessary and sufficient conditions for the re gular projective objects to coincide with the U-projectives. Finally, we delineate how these results nay be applied in certain familiar situations. 相似文献
4.
《Quaestiones Mathematicae》2013,36(3):279-291
ABSTRACT In the general setting of a complete, well-powered category A, we define and study two universal closure operations: regularization and extremalization, by means of regular and extremal subobjects of A. respectively. A general theorem of characterization of epimorphisms in A is given. When A is an epireflective subcategory of TOP, such operations are shown to coincide with A-closure [11] and epiclosure [2]. respectively. In the topological contest, regularization and extremalization are studied in detail and compared with r-closure, defined in [13]. 相似文献
5.
《Quaestiones Mathematicae》2013,36(3-4):265-272
Abstract Nets and graphs, both used in Computer Science, are studied from a categorical point of view. It is shown that they may be constructed via final completions of very simple small concrete categories and that their nice properties, namely to form topological categories which are quasitopoi with concrete powers such that products of final maps are final, depend on this fact. Furthermore, the relations between them can be described by means of adjoint functors. 相似文献
6.
《Quaestiones Mathematicae》2013,36(2):181-196
Abstract Some modules, ?p (B) (1 < p < ∞), over a commutative B*-alge= bra with identity are discussed and these are seen to have properties similar to the Banach spaces ?p (1 < p < ∞). 相似文献
7.
《Quaestiones Mathematicae》2013,36(4):531-547
Abstract For each adjoint functor U: A → X where X is an (?, M)-category having enough ?-projectives, we construct an (?, M)-algebraic hull E: (A, U) → (Â, Û), i.e., (Â, Û) is (epsiv; M)-algebraic and E has a certain denseness property. We show that there is a conglomerate of functors over X with respect to which the (? M)-algebraic categories are exactly the injective objects and characterize (? M)-algebraic hulls as injective hulls. 相似文献
8.
《Quaestiones Mathematicae》2013,36(2):179-201
ABSTRACT Consider an adjunction <F,U;n,c>: K. → A, T = <T,n,u> the monad it induces in K and ø: A → KT the comparison functor, KT being the category of T-algebras. By ø*: Proj U → Proj UT we denote the restriction and co-restriction of ø to the subcategories of U-projective and UT -projective objects, respectively. In this paper we deal with the following problem, raised by R.-E. Hoffmann in [5] 1.16 (b): Assuming that ø* is an equivalence of categories when is it possible to find a category C and a right adjoint functor V: C → K inducing the same monad T in K, and a full reflective embedding E: A → K, such that: (1) V.E = U. (2) ø = ø'. E for the comparison functor ø': C → KT . (3) F'X is contained (via E) in A, for each K-object X, F' being the left adjoint of V. (4) ø': C → KT has a full and faithful left adjoint L'. We prove that there exists a pair (C,V) satisfying the conditions of the problem, with A an isomorphism-closed subcategory of C, such that: (5) For all C ? Obj C the reflection map rC: C → A is ø'-initial. We also prove that this pair (C,V) is the universal solution satisfying condition (5), i.e. if (Ci,Vi) is a pair satisfying conditions (1)-(5) with Ei: A → C2 the embedding and Li left adjoint to the comparison functor øi: Ci KT then there exists a unique full and faithful functor Hi: C → Ci such that H. E = Ei and Hi. L'—Li. Moreover the universal solution is uniquely determined up to isomorphisms of categories and natural isomorphisms of functors. Finally, we study a particular situation and find, within the solutions of the problem satisfying two further conditions, the lease and the largest element. We conclude the paper with an example of this situation. 相似文献
9.
《Quaestiones Mathematicae》2013,36(4):353-362
Abstract In this paper two ordered families of topological categories are studied. The first family includes the category of all abstract simplicial complexes and the subcategories of all abstract simplicial complexes of dimension less than or equal to n. The categories of the second family are bireflective subcategories of the category of all bornological spaces. All these categories are cartesian closed and have other nice properties. 相似文献
10.
《Quaestiones Mathematicae》2013,36(4):475-477
Abstract Using the notion of a spectral set introduced by von-Neumann, we give some characterizations of C+ -algebras. Two conditions, each of which is necessary and sufficient for a Banach algebra to be uniform, are also obtained. 相似文献
11.
《Quaestiones Mathematicae》2013,36(1-3):113-137
Abstract Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:
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if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;
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V carries U-initial monosources into T-initial mono-sources.
12.
《Quaestiones Mathematicae》2013,36(1-3):235-243
A connector U on a space S is a function from S to the power set of S such that each x in s belongs to its image. The image of x is denoted by xU. In other words, the relation {(x,y): y ? xU, x ? S) is a reflexive binary relation. A space with a certain set of connectors is a generalization of topological spaces as well as uniform spaces. In this paper, a notion of completeness of such a space is introduced. This completeness corresponds to completeness of uniform spaces if a set of cannectors meets the conditions of uniformity. Compactness of topological Spaces is a special case of the completeness. 相似文献
13.
《Quaestiones Mathematicae》2013,36(1-3):67-71
Abstract Let K: P → T be a fixed functor. A criterion is given for a functor M': T → V to be a (right) Kan extension along K of some functor M: P → V. The functors M having a given M' as Kan extension are, in general, classified by continuous functors (V P)o → V. We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups. 相似文献
14.
C. D. Godsil 《Combinatorica》1981,1(2):163-167
A graphX is said to beequiarboreal if the number of spanning trees containing a specified edge inX is independent of the choice of edge. We prove that any graph which is a colour class in an association scheme (and thus
any distance regular graph) is equiarboreal. We note that a connected equiarboreal graph withM edges andn vertices has edge-connectivity at leastM/(n−1). 相似文献
15.
We establish a 2-categorical duality involving the 2-category A
of all -accessible categories with wide pullbacks, also known as locally -polypresentable categories, and of functors preserving -filtered colimits and wide pullbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt (A, Set)=A
The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994. 相似文献
16.
We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topological spaces. 相似文献
17.
Manuela Sobral 《Applied Categorical Structures》1996,4(1):97-106
The paper deals with (effective) descent morphisms for subfibrations
X of the basic fibration Top/X, for topological spaces X and classes
of continuous functions stable under pullback. For a category with pullbacks, we prove the stability under pullback of effective
-descent morphisms for a class
satisfying some suitable conditions. This plays a rôle in relating effective
-descent to effective global-descent and enables us to obtain a criterion for effective étale-descent. We also show that the inclusion of the class of effective global-descent maps in the class surjective effective étale-descent is strict.Partial financial support by Centro de Matemática da Universidade de Coimbra is gratefully acknowledged. 相似文献
18.
《Quaestiones Mathematicae》2013,36(4):357-368
Abstract The aim of this paper is to describe some “topological” subclasses of the class of all semi-topological functors. This is done by considering intersections of certain classes: on the one hand, the classes of all semi-topological functors, all topologically-algebraic functors and all regular functors; and on the other hand, the class of all discrete functors (see [6]). The relations between these and other classes of functors are given along with the necessary examples. Finally, the resulting intersections are described by giving representations of the functors that belong to them. 相似文献
19.
For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups. 相似文献
20.
We study those 2-monads on the 2-categoryCat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day.The first author gratefully acknowledges the support of the Australian Research Council. 相似文献