Applied category theory

  • Course teacher Dario Bojanjac
  • Associate teachers -
  • ECTS credits 5
  • Number of hours: Lectures + Seminars + Exercises 45 0 0

Course objectives

The aim of the course is to define basic mathematical concepts such as set, relations, function and order, and to motivate basic concepts of category theory.

Category theory is one of the basic disciplines of theoretical mathematics and in last decades it has become one of the basic frameworks for describing the connections between abstract mathematical concepts.

The strength of theoretical mathematics is in finding analogies between analogies and on this task no other mathematical theory can be compared with the theory of categories. The peculiarity of category theory in relation to other abstract mathematical theories is its direct applicability in the natural and social humanities and in connecting different concepts into one whole.

Enrolment requirements and/or entry competences required for the course

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Learning outcomes at the level of the programme to which the course contributes

  • Apply theoretical knowledge of the fundamentals of the six core disciplines and their relationship within cognitive science.
  • Apply specific knowledge and skills from selected disciplines constituting cognitive science.
  • Integrate insights, methods, and levels of analysis across different disciplines into a unified framework for understanding the human mind and cognition in general.
  • Employ diverse disciplinary tools in exploring and describing the nature of cognitive processes.

Course content (syllabus)

  1. Introduction to mathematical logic and mathematical thinking.
  2. Introductions to sets, relations and functions.
  3. Preorder and monotone maps.
  4. Introduction to Galois connections.
  5. Introduction to resource theories.
  6. Introduction to categories. Isomorphism in a category.
  7. Functor, natural transformations and databases.
  8. Adjunctions and data migration.
  9. Enriched profunctors.
  10. The basic idea of categorification.
  11. Simplified signal flow graphs.
  12. Graphical linear algebra.
  13. Colimits and connection.
  14. Operads and their algebras. Wiring diagrams.
  15. How can we prove our machine is safe?

Student responsibilities

Class attendance. Independent assignments. Midterm exam and final exam.

Required literature

  • T. - D. Bradley, What is Applied Category Theory, arXiv 2018
  • B. Fong, D. I. Spivak, An Invitation to Applied Category Theory: Seven Sketches in Compositionality, Cambridge University Press 2019

Optional literature

  • D. I. Spivak, Category Theory for the Sciences, The MIT Press 2014