not null-homotopic such that [Z,X] [Z, f ] → [Z, Y ] is constant. For any X ′ ⊂ X with fewer than κ cells there exists X ′ → Z → X ′ = 1X′ , from which we may conclude that [X ′, X] [X ′, f ] → [X ′, Y ] is constant and in particular that f |X ′ is null-homotopic. Let H be the homotopy category obtained from T . Its objects are the objects of T , its maps are homotopy-classes of maps. The theor...

We investigate 3-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of E-subtractive varieties (where E is the set of constants in a variety) recently introduced by the f...

Let   a family of monomorphisms in the category A ‎To study mathematical notions in a category  A‎, ‎such as injectivity‎, ‎tensor products‎, ‎flatness‎, ‎with respect to a class M of its (mono)morphisms‎, ‎one should know some of the categorical properties of the pair (A ‎,M ‎)‎. ‎In this paper we take A to be the‎‎category  Pos-S  of S-partially ordered sets  and   to be a particular‎ class o...

We study Baire category for downward-closed subsets of 2ω , showing that it behaves better in this context than for general subsets of 2ω . We show that, in the downward-closed context, the ideal of meager sets is prime and b-complete, while the complementary filter is g-complete. We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results fo...

Let C be a small category and Set the category of sets. We define a C-set to be a functor Ω : C → Set. Thus Ω is simply a diagram of sets, the diagram having the same shape as C: for each object x of C there is specified a set Ω(x) and for each morphism α : x → y there is a mapping of sets Ω(α) : Ω(x) → Ω(y). If C happens to be a group (a category with one object and morphism set G) then a C-se...

This paper first improves Chen and Hsieh’s definition of generalized fuzzy numbers, which makes it the generalization of definition of fuzzy numbers. Secondly, in terms of the generalized fuzzy numbers set, we introduce two different kinds of orders and arithmetic operations and metrics based on the λ-cutting sets or generalized λ-cutting sets, so that the generalized fuzzy numbers are integrat...

By a multi-pointed category we mean a category C equipped with a so called ideal of null morphisms, i.e. a class N of morphisms satisfying f ∈ N ∨ g ∈ N ⇒ fg ∈ N for any composable pair f, g of morphisms. Such categories are precisely the categories enriched in the category of pairs X = (X,N) where X is a set and N is a subset of X, whereas a pointed category has the same enrichment, but restri...