[论文解读] Notes on Category Theory with examples from basic mathematics

Paolo Perrone|arXiv (Cornell University)|Dec 23, 2019

Homotopy and Cohomology in Algebraic Topology参考文献 9被引用 11

一句话总结

本文通过基础数学中的具体例子，以直观、基础的视角介绍了范畴论，阐述了范畴、函子、自然变换和普遍性质的概念。它证明了范畴正是图上某一特定单子的代数，且函子恰好是这些代数的态射，从而将范畴结构与代数单子理论统一起来。

ABSTRACT

These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field. Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as "just the dual to monads". Theorems, interpretations and concrete examples are given for monads as well as for comonads. This work, thoroughly revised and expanded, is now a book, with an extra section on monoidal categories.

研究动机与目标

• 为数学及相关领域的初学者提供一个自包含的、以实例为导向的范畴论入门。
• 通过熟悉的数学结构，澄清范畴、函子、自然变换和普遍性质等基础概念。
• 通过证明范畴正是图上某一单子的代数，建立范畴论与单子理论之间的深层联系。
• 证明范畴之间的函子恰好是这些单子代数的态射，从而统一范畴论与代数视角。

提出的方法

• 使用范畴的多种直观解释：作为关系、运算，以及带有映射的结构化空间。
• 通过在有向图范畴上定义单子 T，使得 T-代数对应于范畴。
• 通过在图中行走路径上编码复合与单位公理，构造单子 T。
• 通过路径复合验证单位与乘法公理，证明 T-代数恰好是小范畴。
• 通过验证涉及单子结构的关键图表的交换性，证明范畴之间的函子恰好对应于 T-代数的态射。
• 利用 Yoneda 引理和普遍性质，将范畴构造建立在可表示性与典范映射的基础上。

实验结果

研究问题

• RQ1 如何仅使用基本数学例子和直观解释来引入范畴论？
• RQ2 范畴论公理背后的代数结构（具体而言，单子）是什么？
• RQ3 当范畴被视为单子的代数时，函子是否恰好是范畴之间的态射？
• RQ4 通过这一单子框架，能否推导并理解积、极限和伴随的普遍性质？
• RQ5 Yoneda 引理和可表示性如何在这种范畴的代数表述中自然出现？

主要发现

• 范畴恰好是图上单子 T 的代数，其中 T 通过路径编码了复合与单位公理。
• 范畴之间的函子恰好是 T-代数的态射，因为它们保持单子结构并使相关图表交换。
• T-代数结构通过单位与乘法公理，构造性地保证了恒等与复合的结合律。
• Yoneda 嵌入与普遍性质在此框架中自然恢复，表明可表示函子通过单子代数结构与对象一一对应。
• 该构造为范畴与函子提供了清晰的代数刻画，统一了范畴论与普遍代数的视角。
• 该结果可推广，表明整个范畴论框架——包括极限、伴随与普遍构造——均可视为单子-代数结构的实例。