用范畴论进行数学建模：模型及其关系
Mathematical modelling by help of category theory: models and relations between them
摘要：
现代实际问题的日益复杂对数学建模提出了更高的要求。由于各种模型可用于模拟一种物理现象，模型比较和模型选择的作用变得特别重要。目前在实际应用中通常使用的模型比较和模型选择方法是基于计算的，因此耗时且计算成本高。因此，有必要开发其他的抽象方法，即不需要计算，而使用数学模型。数学模型的抽象描述可以借助于抽象数学来实现，这意味着模型及其之间关系的形式化。本文提出了一种基于范畴论的数学建模方法。这样，数学模型就用范畴的语言形式化了，模型之间的关系也被形式化地定义了，并且在范畴的层次上引入了一些与实际相关的性质。最后，一个说明性的例子是如何在实践中使用的范畴理论为基础的方法。此外，本文中提出的所有结构也从建模的角度进行了讨论，通过明确的链接到具体的建模方案。
英文摘要：
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.
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