Intro to Quasi-categories

MATH 8850 (UVA)

Course information

  • Lectures: Mondays & Wednesdays, 5:00pm-6:15pm
  • First Lecture: 25. August, 2021
  • Office: 311
  • Office hours: Friday 3:00-5:00pm. Additional office hours by appointment.
  • Syllabus
  • Notes

Exercise Sheets

Exercise sheets will appear here and on the course collab page. A “Homework 0”, which will assess students’ categorical background, will be added at the end of July, 2021. Students intending to take the course should attempt “Homework 0”, and email it to me before the beginning of the course, as the content of the first two weeks will be determined by the responses.


In the past fifty years, various versions of higher categories have become common in many branches of mathematics. Fukaya categories in symplectic geometry, derived/higher stacks in algebraic geometry, and (E_n)-algebras in homotopy theory (to name just three examples) all require some kind of higher-categorical “coherent” language to formulate and work with. As higher-categorical methods have proliferated, Joyal’s theory of quasi-categories has enjoyed particularly rapid development and great popularity in recent decades, in large part due to the work of Lurie in extending this theory and connecting it to algebraic geometry.

The aim of this course is to provide an introduction to the theory of quasi-categories, up to a point where students will be able to access the burgeoning literature making use of quasi-categorical techniques. While a number of monographs on quasi-category theory exist, these are often highly technical and short on intuitive explanation. This course will therefore aim at providing both rigor and intuition for the fundamentals of quasi-category theory.

Content to be covered in this course will include:

  • Simplicial sets — both as models for spaces, and as models for higher categories.
  • Some background on model categories.
  • The Kan-Quillen, Joyal, and Bergner model structures, and some relations between them.
  • Basic constructions in quasi-category theory.

Time permitting, we will discuss some of the following topics:

  • Limits and colimits in quasi-categories
  • The Grothendieck construction
  • Monoidal ((\infty,1))-categories.


Some background knowledge will be necessary to follow the course. In particular

  • You should have completed Algebraic Topology I (or an equivalent) prior to the beginning of the course.
  • You should be familiar with the basic constructions of category theory: categories, functors, natural transformations, limits, colimits, Kan extensions, and adjunctions. (If your background in category is a bit shakier, see the “References” section.)

A few weeks before the start of the semester, I will provide a “Homework 0” to gauge your categorical background. The content of the first few weeks may change based on student responses to Homework 0.

Taking Algebraic Topology II either prior to or at the same time as this course could be useful, but is not necessary. We will occasionally use more advanced facts from algebraic topology as black-boxes during the course.


There will be weekly exercise sheets, and students will be expected to hand in solutions to at least 60% of the exercises assigned. The final grades for the course will be decided based on one-on-one discussions of the material at the end of the semester.


For those who are less conversant with category theory the following references may be of help. I particularly enjoy the first, both for its writing style and wealth of examples.

  • Riehl, E. 2016: Category Theory in Context. Dover. 258 pp. Available online
  • Leinster, T. 2014; Basic Category Theory. Cambridge University Press. 191 pp. arXiv:1612.09375

The primary reference for the course will be a set of typed lecture notes I will provide. However, the following works may be useful references during the course.

  • Bergner, J. 2018: The Homotopy Theory of ((\infty,1))-Categories. Cambridge University Press. 284 pp.
  • Cisinski, D.-C. 2020: Higher Categories and Homotopical Algebra. Cambridge University Press, 446 pp. (Available online)
  • Goerss, P. and Jardine, R. 1999: Simplicial Homotopy Theory. Birkhäuser, 510 pp.
  • Groth, M. 2015: A short course on (\infty)-categories. 77 pp. (arXiv:1007.2925)
  • Hovey, M. 1999: Model Categories. American Mathematical Society. 209 pp.
  • Joyal, A. 2008: Notes on Quasi-Categories. 244 pp. (Available online)
  • Lurie, J. 2009: Higher Topos Theory. Princeton University Press. 944 pp. (Available online)
  • Riehl, E. 2014: Categorical Homotopy Theory. Cambridge University Press. 292 pp. (Available online)