Approach to formalization of the intelligent information control systems based on the topos theory

The approach to formalization of complex organizational and technical systems based on the toposes theory is substantiated. The article presents a review of the literature on the formalization of knowledge. An integrated control system for technical complexes is considered. The description of blocks of intelligent information and control systems based on the toposes theory is developed. An algorithm for formalizing intelligent information and control systems and an example of constructing toposes and functors for the general case are presented. The conclusion is made about the possibility of applying the proposed approach to the formalization of the intelligent information and control systems in the process of formation of their appearance.

The Scott model of PCF in univalent type theory

We develop the Scott model of the programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for the non-termination in PCF, we use the lifting monad (also known as the partial map classifier monad) from topos theory, which has been extended to univalent type theory by Escardó and Knapp. Our results show that lifting is a viable approach to partiality in univalent type theory. Moreover, we show that the Scott model can be constructed in a predicative and constructive setting. Other approaches to partiality either require some form of choice or quotient inductive-inductive types. We show that one can do without these extensions.

Knowledge, Fear, and Conscience: Reasons to Stop Flying Because of Climate Change

Much research on the societal consequences of climate change has focused on inaction, seeking to explain why societies and individuals do not change according to experts’ recommendations. In this qualitative study, we instead consider people who have changed their behaviour for the sake of the climate: They have stopped travelling by air. We first asked them to elaborate their rationales for the behaviour change. Then, using topos theory to find thought structures, we analysed their 673 open-text answers. Several themes emerged, which together can be regarded as a process of change. Increased knowledge, primarily narrated as a process by which latent knowledge was transformed into insight, through experience or emotional distress, was important. Contrary to certain claims in the literature, fear stimulated change of behaviour for many in this group. Climate change was framed as a moral issue, requiring acts of conscience. Children were invoked as educators and moral guides. Role models and a supportive social context played an important part. Alternatives to flying were brought forward as a motive to refrain from flying. Only a few mentioned shame as momentous. Instead, stopping travelling by air invoked a feeling of agency and responsibility, and could also result in a positive sensation.

Innovative Design Within Tradition - Injecting Topos Structures in C-K Theory to Model Culinary Creation Heritage

In “Grande cuisine” creation and tradition co-evolve in a rich number of ways. Great chefs still use recipes from the 19th century and may also reinvent gastronomy itself. The creation heritage of culinary Art is the paradoxical capacity to both “respect” tradition and “break” its rules. Building on C-K theory, we show that such creative heritage needs multiple and independent layers of knowledge that “speak” of basic fixed objects. These properties correspond to general mathematical structures that we find in Topos theory. Thus, C-K/Topos predicts creative design strategies that can respect tradition in different ways. It also proves a form of “innovation within tradition” - “sheafification” in Topos words- that is not a compromise and builds on tradition itself. These findings fit with the lessons of great books of gastronomy. C-K/Topos has a wide scope of validity: it applies to any innovative design that needs preserving systemic structures, like engineering systems or social and environmental systems. C- K/Topos models with a high generality how local and radical innovation can warrant systems incremental change. C-K/Topos will have implications for teaching and research.

Grothendieck Topoi: Architectural and Plastic Imagination beyond Material Number and Space

Fernando Zalamea begins with an emphatic argument that from mathematics to its plastic influence, Grothendieck topoi are used to understand the human and the nonhuman creativity in three variously relevant and yet different disciplines; that of cinema through the work of Andrei Tarkovsky, architecture through the work of Frank Gehry and art through the work of Anselm Hiefer. The chapter starts by introducing the work of Alexandre Grothendieck from a conceptual standpoint, focusing on his topos theory (1962), a general setting which encompasses both arithmetic (number) and geometry (space) under a common abstract perspective (sheaves: a far-reaching tool which helps to glue the local and the global). Grothendieck's revolution, wider than Einstein's interlacing of space and time, has radically changed mathematics, but its plastic influence beyond the specialty has yet to be developed.