Categorical foundations of variety-based topology and topological systems

Topological systems and Artin glueing

“…To provide the motivation for the study of Artin glueing we begin the paper recalling some definitions and results presented in [60]. In particular, this section contains the algebraic preliminaries indispensable for understanding the new notions.…”

“…Everything is now on its place to introduce the promised variety-based topological theories [60]. Recall from Introduction that we are making a mixture of powerset theories of [53,Definition 3.5] and topological theories of [1, Exercise 22B] as well as using our previous paper [67].…”

“…Given a subcategory C of LoA, the restriction Set×C The reader should notice that the variety of closure lattices introduced in [60] provides a way for including the case of closure spaces [5,6]. Moreover, an important property of the category Top(T ) is contained in the next theorem, whose proof is based on a generalization of the standard result of classical (and fuzzy) topology that continuity of a map can be checked on elements of a subbasis.…”

“…The real reason for so doing was given quite recently in [60,61], where we showed that a particular subcategory of Top(T ) provides a framework for the concept of state property system introduced by D. Aerts [4] to serve as the basic mathematical structure in the GenevaBrussels approach to foundations of physics (the structure was further developed into the notion of state context property system [7,8] providing a framework for studying concepts arising in human mind; the approach can model an arbitrary combination of concepts by making use of the standard quantum mechanical procedure to describe the combinations of quantum entities). This example reveals the true power of our approach, showing that it is capable of uniting concepts from quite different areas of nowadays science (physics is certainly not a subdivision of mathematics).…”

“…To fill the gap we introduced in [60] variety-based topological theories making a mixture of powerset theories of [53,Definition 3.5] and topological theories of [1, Exercise 22B] as well as using our previous results in the field [67]. With the help of the new approach we obtained a category of topological spaces which appeared to be topological over its ground category (an analogue of [53,Theorem 3.10]).…”