Mathematical modelling by help of category theory: models and relations between them

TL;DR

This paper proposes a category theory-based framework for abstract mathematical modelling, enabling formal comparison and relation of models without computational methods, illustrated through practical examples.

Contribution

It introduces a novel category theory approach to formalize models and their relations, enhancing abstract analysis and comparison of mathematical models.

Findings

Formalization of models using categories

Definition of relations between models in categorical terms

Practical example demonstrating the approach

Abstract

The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus, time consuming and computationally costly. Therefore, it is necessary to develop other approaches for working abstractly, i.e. without computations, with mathematical models. The abstract description of mathematical models can be achieved by help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. On this way, mathematical models are formalised in the language of categories, relations between the models are formally defined, as well as several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from the modelling point of view by making explicit the link to concrete modelling scenarios.