In this paper, we introduce a subcategory ∼Sh* of Sh* and obtain some results in this subcategory. First we show that there is a natural bijection Sh(∑(X, x), (Y,y))≅Sh((X,x),Sh((I, Ī),(Y,y))), for every (Y,y)∈ ~Sh* and (X,x)∈ Sh*. By this fact, we prove that for any pointed topological space (X,x) in ∼Sh*,πntop(X,x)≅ πn-ktop(Sh((Sk, *),(X,x)), ex), for all 1≤k ≤n-1.

In this paper first we define the morphism between geometric spaces in two different types. We construct two categories of U-GESP and L-GESP from geometric spaces then investigate some properties of the two categories, for instance U-GESP is topological. The relation between hypergroups and geometric spaces is studied. By constructing the category SNS-Hv of Hv-groups we answer the question that which construction of hyperstructures on the category of sets has free object in the sense of universal property. At the end we define the category of geometric Hypergroups and we study its relation with the category of hypergroup.

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The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. We introduce R−, basis and R−, solution on rigid objects of a monoidal 𝐴, 𝑏, −, category, for a compatibility relation R, such that we require the notion of duality introduced by Joyal and Street, the notion given by Yetter and Freyd and the classical one, then we show that R−, solutions over a semisimple ribbon 𝐴, 𝑏, −, category form as well a semisimple ribbon 𝐴, 𝑏, −, category. This allows us to define a concept of so-called quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.

Several fuzzy topologies are de ned and studied by di erent authors. In this article, we unify  ve of the most common fuzzy topologies existing in the literature, as well as the standard topology. This is done by introducing the notion of structural topology on objects in a category and proving that topologies on a set as well as fuzzy topologies on fuzzy sets and fuzzy topologies on fuzzy subsets are all structural topologies. We also introduce the notion of structural continuity and we show that the fuzzy continuity de ned in the literature in all the above mentioned cases, as well as the standard continuity are structural.

For many centuries the linguists have almost always studied morphology. But with the advent of Transformational Grammar , syntactic studies and the structure of sentences have earned maximum importance, therefore Sentence was the focus of study until 70s , and from then on , little by little linguists have begun to study the structure of NP, and they have found many similarities between NPs , VPs and sentences . The idea of this similarity has led to many comparative studies. The Empty category is one of the arguments discussed in sentence structure and pro /PRO is one of these empty categories. This article intends to indicate the existence of pro/PRO in NPs . Comparing Persian and Italian languages we find the same control properties in sentences.