adjunction

简明释义

英[əˈdʒʌŋkʃ(ə)n]美[əˈdʒʌŋkʃən]

n. 添加；附益

英英释义

单词用法

同义词

反义词

例句

1.As an efficient instructional technology of reading, adjunction can facilitate reading comprehension.

附加问题是一种有效的阅读教学技术。

2.This story thus leaves us in the three, since the adjunction of the four is superfluous.

这个故事因此让我们留在这三个环结。因为这四个环结的添加是多余的。

3.As an efficient instructional technology of reading, adjunction can facilitate reading comprehension.

附加问题是一种有效的阅读教学技术。

4.The adjunction between the free functor and the forgetful functor is a fundamental aspect of algebra.

自由函子和遗忘函子之间的伴随是代数的一个基本方面。

5.The notion of adjunction plays a crucial role in the development of homotopy theory.

在同伦理论的发展中，伴随的概念起着至关重要的作用。

6.In functional programming, the concept of adjunction helps in understanding type classes.

在函数式编程中，伴随的概念有助于理解类型类。

7.Understanding adjunction can help clarify many concepts in both mathematics and computer science.

理解伴随可以帮助澄清数学和计算机科学中的许多概念。

8.In category theory, an important concept is the adjunction, which describes a relationship between two functors.

在范畴论中，一个重要的概念是伴随，它描述了两个函子的关系。

作文

In mathematics, particularly in category theory, the concept of adjunction plays a crucial role in understanding the relationships between different categories. An adjunction consists of two functors that are linked together in a way that provides a deep insight into their structure and behavior. To illustrate this, let us consider two categories, A and B, and two functors, F from A to B and G from B to A. The functor F is said to be left adjoint to G if there is a natural isomorphism between the hom-sets, which can be expressed as Hom_B(F(a), b) ≅ Hom_A(a, G(b)) for every object a in A and b in B. This relationship signifies that F preserves certain limits while G reflects them, creating a harmonious interaction between the two categories.The importance of adjunction extends beyond pure mathematics; it has applications in various fields such as computer science, particularly in type theory and programming language semantics. For instance, in functional programming, the notion of adjunction can be observed in the relationship between the types of data structures and their respective operations. When we define a data type, we often provide a way to construct it and a way to deconstruct it, which can be seen as a pair of adjoint functors. This duality allows programmers to reason about the transformations of data in a more structured manner, ensuring that operations can be performed safely and effectively.Moreover, the concept of adjunction can also be applied in the realm of logic. In many logical systems, especially those that deal with modal logic or intuitionistic logic, the idea of adjunction helps clarify how different logical operators interact with each other. For example, the relationship between necessity and possibility can be framed within an adjunction, providing a clearer understanding of how these modalities relate to one another. By examining the adjunction, logicians can derive new insights into the nature of logical entailment and the structure of logical arguments.In summary, the notion of adjunction is a powerful tool that bridges various domains of study, facilitating a deeper understanding of complex relationships between structures. Whether in mathematics, computer science, or logic, adjunction serves as a unifying concept that enhances our comprehension of how different entities interact and relate to one another. As we continue to explore the implications of adjunction, we uncover new pathways for innovation and discovery across disciplines, highlighting the interconnectedness of knowledge and the beauty of mathematical abstraction.In conclusion, grasping the essence of adjunction not only enriches our understanding of theoretical constructs but also empowers us to apply these ideas practically in various fields. The exploration of adjunction reveals the intricate tapestry of relationships that exist within mathematics and beyond, encouraging us to think critically and creatively about the world around us.

在数学中，特别是在范畴论中，adjunction的概念在理解不同范畴之间的关系中起着至关重要的作用。adjunction由两个范子组成，它们以一种方式相互关联，提供了对其结构和行为的深刻洞察。为了说明这一点，让我们考虑两个范畴A和B，以及从A到B的两个范子F和从B到A的范子G。如果存在一个自然同构，表示为Hom_B(F(a), b) ≅ Hom_A(a, G(b))，则范子F被称为G的左伴随。这种关系意味着F保持某些极限，而G反映它们，创造了两个范畴之间的和谐互动。adjunction的重要性超越了纯数学；它在计算机科学等多个领域具有应用，特别是在类型理论和编程语言语义学中。例如，在函数式编程中，adjunction的概念可以在数据结构的类型与其各自操作之间的关系中观察到。当我们定义一种数据类型时，我们通常会提供一种构造它的方法和一种解构它的方法，这可以看作是一对伴随的范子。这种对偶性使程序员能够以更结构化的方式推理数据的转换，确保操作可以安全有效地执行。此外，adjunction的概念也可以应用于逻辑领域。在许多逻辑系统中，特别是那些处理模态逻辑或直觉逻辑的系统中，adjunction的思想有助于阐明不同逻辑运算符之间的相互作用。例如，必要性和可能性之间的关系可以在一个adjunction中框架，从而提供对这些模态如何相互关联的更清晰理解。通过检查adjunction，逻辑学家可以得出关于逻辑蕴涵性质和逻辑论证结构的新见解。总之，adjunction的概念是一个强大的工具，桥接了各种研究领域，促进了对复杂结构之间关系的更深入理解。无论是在数学、计算机科学还是逻辑中，adjunction作为一个统一的概念，增强了我们对不同实体如何相互作用和相关的理解。随着我们继续探索adjunction的含义，我们发现了跨学科创新和发现的新路径，突显了知识的相互联系以及数学抽象的美。最后，掌握adjunction的本质不仅丰富了我们对理论构造的理解，还使我们能够在各个领域实际应用这些思想。对adjunction的探索揭示了存在于数学及其他领域的复杂关系的精致织锦，鼓励我们批判性和创造性地思考我们周围的世界。