Introduction

You just looked at yet another introduction to Category Theory. The subject mostly consists of a lot of definitions that are related to each other. We wrote the book to collect all the definitions in one place to be checked and updated easily in the future when we decide to refresh our knowledge about the field of mathematics. Thus the book was written mostly for our category theory study purposes, but we shall appreciate it if somebody else finds it useful.

The topics (chapters) cover base definitions (Object, Morphism and Category) as well as more advanced ones (Functor, Natural transformation, Monad) and also include important results from category theory such as Yoneda’s lemma (see Yoneda’s lemma) and Curry-Howard-Lambek correspondence (see Curry-Howard-Lambek correspondence). The Topos gives an introduction to topos theory i.e. just another view of Sets.

There are a lot of examples in each chapter. The examples cover different category theory application areas. We assume that the reader is familiar with the corresponding area and the examples can be skipped otherwise. I.e. anyone can choose suitable examples for themselves.

The most important examples are related to the set theory. The set theory and category theory are very closely related. Each one can be considered as an alternative view of the other one.

There are also several examples from programming languages which include Haskell, Scala, C++. The source files for programming languages examples (Haskell, C++, Scala) can be found on github repositories:

The examples from physics are related to quantum mechanics that is the best known to us. For the examples we were inspired by the Bob Coecke article (Coecke 2008).

There is also additional material related to abstract algebra (see Abstract algebra) taken from (Murashko 2016-2019). The material describes the different math constructions used in the book.

The text is distributed under Creative Common Public License (see the text of the license at the end of the book) i.e. any reader has the right to copy, store, modify, distribute or build upon the book as long as the original authors are mentioned in the derivative products. The initial text of the book can be found at (Murashko 2018).

Coecke, Bob. 2008. “Introducing Categories to the Practicing Physicist.” https://arxiv.org/abs/0808.1032.
Ivan Murashko, Marat Minshin, Alexey Radkov. 2018a. “C++ Examples for Category Theory by Example Book.” GitHub Repository. GitHub. https://github.com/CatTheoryByExample/cpp-examples.
———. 2018b. “Haskell Examples for Category Theory by Example Book.” GitHub Repository. GitHub. https://github.com/CatTheoryByExample/hs-examples.
———. 2018c. “Scala Examples for Category Theory by Example Book.” GitHub Repository. GitHub. https://github.com/CatTheoryByExample/scala-examples.
Murashko, Ivan. 2016-2019. “Lectures Notes in Introduction to Galois Theory by Ekaterina Amerik.” GitHub Repository. GitHub. https://github.com/ivanmurashko/courseragalois.
———. 2018. “Category Theory by Example.” GitHub Repository. GitHub. https://github.com/ivanmurashko/articles/tree/master/cattheory.