Category theory and simplicial methods

 MA 598- Spring 2023, Prof. Manuel Rivera,  Purdue University

Course description: This course is an introduction to category theory with emphasis on simplicial methods and their use in topology, algebra, and algebraic geometry. In the first part, we will develop the basics of category theory together with many examples. In the second part, we will focus on homotopy theory and simplicial methods with a view towards higher category theory. This course should be accessible to a relatively wide range of graduate students with different background and interests.

Background: Familiarity with basic algebraic topology will be helpful. 


Part 1: Basic category theory through examples

  • Categories, Functors, and Natural Transformations
  • Universal properties and (co)limits
  • Adjunctions and equivalences
  • Monoidal categories
  • Monads an algebras

Part 2: Homotopy theory and simplicial methods

  • Classical homotopy theory
  • Weak equivalences and localization
  • Kan extensions and derived functors
  • Homotopy (co)limits
  • Abstract homotopy theory and model categories


We will collectively write and edit a set of notes. Each registered student will be in charge of typing one or two lectures, filling details of arguments outlined in class, and typing solutions to exercises and examples. This will be done on a .tex file that will be shared with the whole class. Collaboration is encouraged. We will continuously keep editing the set of notes during the semester.

Here are some references. For the first part of the course, two good references are Category theory in context by Riehl, freely available here, and the classical text Categories for the working mathematician by Mac Lane. We will complement the topics discussed in these books with many other examples. Some references for the second part of the course are Categorical homotopy theory by Riehl, freely available here, Simplicial homotopy theory by Goerss and Jardine, and Model categories by Hovey. Other useful references are A Concise Course in Algebraic Topology by May and Methods of Homological Algebra by Gelfand and Manin.

Course schedule: Tuesday / Thursday 9:00-10:15 am at Helen Schleman Hall 309

Office hours: TBD

Grading: 100% homework (typing lectures and filling details to arguments and exercises)

Special accommodations: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: or by phone: 765-494-1247.

In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: