2019 Topics in Algebra

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Academic unit or major
Graduate major in Mathematical and Computing Science
Tsuchioka Shunsuke 
Course component(s)
Day/Period(Room No.)
Mon7-8(W832)  Thr7-8(W832)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
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Course description and aims

Give an introductory course on category theory

Student learning outcomes

Get understand the universal property via adjoints, representabilities and limits.


category, functor, natural transformation, adjoint, triangle identity, universality, representability, Yoneda lemma, limit, create limit

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Give lectures on the textbook using blackboard

Course schedule/Required learning

  Course schedule Required learning
Class 1 categories, functors and natural transformations Understand the definitions of cateories, functors and natural transformations
Class 2 adjoints and unit/counit. Understand the definitions of adjoints and characterization via triangle identities
Class 3 universalities Understand relationships between universalities and adjoints
Class 4 representabilities and Yoneda lemma Understand the definition of representabilities and representations of functors by Yoneda lemma (universal elements)
Class 5 limits and colimits Understand the definitions of limits and colimits
Class 6 limits and functors Understand interactions between limits and functors
Class 7 midterm exam evaluate your achievement
Class 8 limits and representabilities Understand relationships between limits and representabilities
Class 9 limits and adjoints Understand relationships between limits and adjoints
Class 10 limits in functor categories Understand behaviour of limits in functor categories
Class 11 adjoint functor theorem Understand the general adjoint functor theorem and applications
Class 12 topos Understand elementary toposes
Class 13 final exam evaluate your achievement
Class 14 computer science and category theory Explain examples of how categories are used in computer science
Class 15 recent topics on categorification Explain examples of categorifications in mathematics


Basic Category Theory, Tom Leinster, Cambridge Studies in Advanced Mathematics (avaiable at https://arxiv.org/abs/1612.09375)

Reference books, course materials, etc.

I will open a lecture webpage and upload supplementing materials

Assessment criteria and methods

based on a relative evaluation of midterm and final examinations

Related courses

  • MTH.A301 : Algebra I
  • MTH.A302 : Algebra II
  • MTH.A331 : Algebra III
  • MCS.T201 : Set and Topology I
  • MCS.T202 : Exercises in Set and Topology I
  • MCS.T221 : Set and Topology II
  • MCS.T222 : Exercises in Set and Topology II
  • ZUA.B201 : Set and Topology I
  • ZUA.B202 : Exercises in Set and Topology
  • ZUA.B203 : Set and Topology II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

basic knowledge of algebra will be desirable

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