Let λ be an infinite regular cardinal and A a locally λ-presentable additive category. We show that any λ-pure morphism (resp. λ-pure quotient) in A creates a kernel-cokernel pair. This implies that the class of all λ-pure kernel-cokernel pairs in A forms an exact structure. Additionally, we will describe λ-pure kernel-cokernel pairs in A and will prove that any λ-directed diagram of objects in A induces a canonical λ-pure kernel-cokernel pair.

Let L be the category of all locally compact abelian (LCA) groups. In this paper, the groups G in L are determined such that every extension 0®X®Y®G®0 with divisible, s-compact X in L splits. We also determine the discrete or compactly generated LCA groups H such that every pure extension 0®H®Y®X®0 splits for each divisible group X in L.

We construct a q-model structure, an h-model structure and an m-model structure on multipointed d-spaces and on flows. The two q-model structures are combinatorial and left determined and they coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed d-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved than the q-model structures.

This paper introduces an approach to the axiomatic theory of quadratic forms based on presentable partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of quadratically presentable fields, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields.

Let à be the category of totally disconnected, locally compact abelian groups. In this paper, we determine the discrete or compact pure injective groups in Ã. Also, we determine the compact pure projective groups in Ã.

Undoubtedly, the design and establishment of Masjid-ol-Haram and Ka’abah, the Qiblah of the Muslim people, are among the simple yet complicated architectures. Simple in the sense that it presents a clear communicative space and complicated in that it is based on the verses of the holy Quran including its place of establishment, the pioneer architect, and its overall structure. Therefore, it is believed it will have the most spiritual influence on the pilgrims.This article, searching through the Quranic verses and the narratives, aims to seek for the recognition of God’s specific will in the essence and idea of the Ka’abah and also attempts to understand how its constructors had built its shape from form. In doing so, a variety of recognitions would be scrutinized as the major part of man’s verity in perceiving that sacred place.Further along, it is attempted to show the perceived truth about the form of the architecture of the Ka’abah in correspondence with the cases of temporal and local brought forth by the scientific intellect achieved in other buildings; so that is this way the structure and presence of this holy monument and the appreciation of the architect can be better understood; therefore, it becomes necessary to have a qualitative focus on other significant disciplines such as ontology, episteme and science.Objectives: 1- Discovering the presentable perspective of the Ka’abah and its relation to the holy verses of Quran.2- Understanding the influence of the history of Ka’abah on its architectural structure and its presentable appearance. Questions: 1- What is the relationship between the intellectual and presentable apprehension of human beings and the structure of the Ka’abah?2- How has the intellectual and presentable architecture of the Ka’abah influenced its overall structure and other architectural monuments?

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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