5M: Category Theory MATHS5079

  • Academic Session: 2018-19
  • School: School of Mathematics and Statistics
  • Credits: 20
  • Level: Level 5 (SCQF level 11)
  • Typically Offered: Semester 1
  • Available to Visiting Students: No
  • Available to Erasmus Students: No

Short Description

Category theory is the study of abstract systems of objects and mappings. As such it is highly general, and it has become indispensable in many parts of modern mathematics, including algebra, topology, logic, and theoretical computer science.

Timetable

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Requirements of Entry

Mandatory Entry Requirements

There are no formal prerequisites. However, many examples will be taken from abstract algebra and topology so an acquaintance with those areas would be an advantage.

Recommended Entry Requirements

Assessment

Assessment

100% Final Exam

 

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

Course Aims

Category theory is the study of abstract systems of objects and mappings. As such it is highly general, and it has become indispensable in many parts of modern mathematics, including algebra, topology, logic, and theoretical computer science.

The main aim of the course is to show how categorical language can simplify mathematical thought and unify apparently unrelated areas of mathematics. Students will learn this language and see examples of how it is used in various other fields, especially algebra and topology. They will also gain technical proficiency in manipulating categorical concepts.

Intended Learning Outcomes of Course

By the end of this course students will be able to:

1. Define category, functor and natural transformation, and give basic examples.
2. Define adjunctions in three different ways (via hom-sets, unit and counit, and comma categories), and prove their equivalence. State and apply the General and Special Adjoint Functor Theorems.
3. Define representable functors. State and prove the Yoneda Lemma and its corollaries.
4. Define and manipulate limits and colimits in general, and specific types such as (co)products and (co)equalizers. Prove that limits can be constructed from products and equalizers. Prove that representables and right adjoints preserve limits. Prove that every presheaf is a colimit of representables. Define cartesian closed category and give examples.

Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.