(1) find colim-acyclic objects in Ab. Here, Ab denote the category of abelian groups and Ab denote the (abelian) functor category for the small category C. The functor colim : Ab → Ab is the direct limit functor and F ∈ Ab is colim-acyclic if colimi F = 0 for i ≥ 1 (see [16] and [5], and the classical books of Cartan and Eilenberg [2] and of MacLane [13]). It is clear that if F is projective th...

We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F : L ✲K i...

The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified, Treumann, based on unpublished ideas of MacPherson, constructed an ‘exit category’ (in the terminology of this paper, the ‘fundamental category’) which classifies constructible sheaves, equivalently stratified etale covers. This pap...

The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample and weakly E-ample semigroups. A semigroup in any of ...

We characterize monomorphisms in rLLoc, the category of regular Lindelöf locales. Though somewhat complicated, the characterization is intrinsic in the sense that it refers only to the properties of the morphism itself, rather than to properties of some lifting of it to a distant category. Monomorphisms in rLLoc cannot be said to be well understood, despite their characterizations in [2] and [7...

UNLABELLED Pain management outcomes assessment depends on valid measurement of pain. However, the validity of single-item scales, such as numeric or faces scales, with the assignment of ordinal numerical values to response scale categories, is questionable. The universal assumption that equal distances between response choices represent equal distances on the dimension being measured is essenti...

fuzzy bi-ideals play an important role in the study of ordered semigroupstructures. the purpose of this paper is to initiate and study theintiuitionistic fuzzy bi-ideals in ordered semigroups and investigate thebasic theorem of intuitionistic fuzzy bi-ideals. to provide thecharacterizations of regular ordered semigroups in terms of intuitionisticfuzzy bi-ideals and to discuss the relationships ...

in this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) of an ordered group (resp. lattice-ordered group) is introduced and some properties, characterizations and related results are given. also, the fuzzy convex subgroup (resp. fuzzy convex lattice-ordered subgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. furthermore,...

In this paper, we describe the Grothendieck group $$K_0(V)$$ of an absolute matrix order unit space V. For purpose, discuss direct limit spaces. We show that $$K_0$$ is a functor from category spaces with morphisms as unital completely $$\vert \cdot \vert$$ -preserving maps to abelian groups. study structure in and prove under certain condition, ordered group. also additive on orthogonal maps.