Applied Category Theory for Engineering II

Abstract

An introduction to Applied Category Theory at a more advanced level than Part I (this "Part II" is self-contained however). Category theory is increasingly used across a broad range of disciplines, from AI to physics, from linguistics to epidemiology. This course teaches the core mathematical concepts and highlights applications in applied sciences, with a special emphasis on engineering.

Objective

Overall goal: to acquire a basic fluency and ability in category theory, and to understand some examples of how it is applied in engineering and adjacent fields. More specifically, by the end of this course, you will be able to: 1) Recognize, name, differentiate, and connect core concepts of category theory, and be able to use them in proofs and calculations. 1a) State from memory the most important definitions/concepts. 1b) Verify by proof/calculation that a given example fits the definition of a concept X. Perform calculations needed for using a mathematical concept for applications. 1c) State which concepts are special cases of other concepts, and how. Compare and contrast mathematical concepts. 2) Describe and illustrate some principle themes and questions of the “compositional way of thinking” as a methodological approach. 2a) Name some concrete examples where "compositionality" plays a role. 2b) List a number of questions that the topic of "compositionality" addresses. 3) Name and describe a few example cases of category theory being used in applied settings. 3a) Name the mathematical structure involved and how it relates to the application in question. 3b) Explain how general principles come to bear in the given examples.

Content

We will cover the follow topics, and connect them to applications across various disciplines: - Categories - Functors - Universal constructions - Natural transformations, Yoneda - Adjunctions - Monads - Comonads - Monoidal categories - Traced monoidal categories - Bicategories

Resources