A Model of Directed Graph Cofiber

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective.

Elements of ∞-Category Theory

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Developing a Spatial Framework for Ship Vulnerability Assessment Based on Category Theory

Ships vulnerability analysis is one of the most important issues in today’s research, to reduce damage and increase safety. To increase the safety of ships, the effective parameters of the vulnerability of ships, the impact of each of them, and the relationship between these parameters should be identified to formulate different scenarios to analyze the vulnerability of ships. This process leads to the formation of simulation models to assess the risk of vessels. The creation of a spatial conceptual framework is needed to create integrated vulnerability models. The most important innovation of this research is the development of a spatial framework for analyzing ships’ vulnerability based on category theory. A framework that can be used to model the various scenarios of ships’ vulnerability from a variety of perspectives. To provide this framework, objects, operators, relationships, and assumptions for vulnerability analysis have been developed. To better express and convey the concepts, the spatial framework of the vulnerability analysis is presented in the form of category theory, which is a mathematical structure. The category theory is a good tool for expressing and creating a mathematical structure for objects and complex relationships in the real world, where other tools do not have this ability. The benefits of the built-in framework have been described with an integrated, precise mathematical structure that can be generalized to other subjects. Studies show that the developed framework is capable of modeling different scenarios for vulnerability analysis to find the best solution to reduce vulnerability.

Knowledge capitalization in mechatronic collaborative design

In mechatronic collaborative design, there is a synergic integration of several expert domains, where heterogeneous knowledge needs to be shared. To address this challenge, ontology-based approaches are proposed as a solution to overtake this heterogeneity. However, dynamic exchange between design teams is overlooked. Consequently, parametric-based approaches are developed to use constraints and parameters consistently during collaborative design. The most valuable knowledge that needs to be capitalized, which we call crucial knowledge, is identified with informal solutions. Thus, a formal identification and extraction is required. In this paper, we propose a new methodology to formalize the interconnection between stakeholders and facilitate the extraction and capitalization of crucial knowledge during the collaboration, based on the mathematical theory ‘Category Theory’ (CT). Firstly, we present an overview of most used methods for crucial knowledge identification in the context of collaborative design as well as a brief review of CT basic concepts. Secondly, we propose a methodology to formally extract crucial knowledge based on some fundamental concepts of category theory. Finally, a case study is considered to validate the proposed methodology.

Semigroups for flows on limits of graphs

We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study ows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory