NB: The official webpage for the course is on the internal Canvas system of the University of Amsterdam (only accessible with UvA login).

Course overview

Category theory is a wonderful tool to understand mathematical structures. Since its origins in algebraic topology in the 1940’s, it has become a unifying language for many fields of mathematics. A category records the relations between mathematical objects by specifying the morphisms between them.

There are nevertheless interesting mathematical phenomena (for instance in algebraic topology, homological algebra, algebraic and differential geometry, mathematical physics, etc.) which category theory does not capture well. The basic problem, which has been understood almost since the beginning of category theory, is that we often need to record finer relationships between morphisms than just equality. A central example of such a finer relation is homotopy between continuous maps of topological spaces (or morphisms of chain complexes).

Infinity-category theory is a modern solution to this old problem. Roughly speaking, in an infinity-category, there is a space of morphisms between any two objects, considered up to coherent homotopy. The theory provides a framework which unifies classical category theory, homotopy theory and homological algebra, and which has already found many applications throughout mathematics.

This course will explain the basic theory of infinity-categories, following an approach based on simplicial sets developed by André Joyal and Jacob Lurie. The main point is that almost all the fundamental constructions of category theory ( limits and colimits, adjoint functors, representable functors and the Yoneda lemma, etc.) can be adapted to infinity-categories.

Recommended prior knowledge: Category theory: categories, functors, natural transformations, representable functors, Yoneda lemma, limits and colimits, adjoint functors. Basic algebraic topology (as in the Master Math course Algebraic Topology I).


The lectures will take place on Zoom.

Time: Wednesday 9:00-11:00

Sessions and course notes:

  • Wednesday September 2nd: Presheaves, simplicial sets. Unfortunately, the notes for this class have been lost. It covered the basic theory of presheaves on small categories and introduced simplicial sets as a special case.

  • Wednesday September 9th: Simplicial sets II. Notes

  • Wednesday September 16th: Infinity-categories. Notes

  • Wednesday September 23rd: Infinity-categories II. Notes

  • Wednesday September 30th: Infinity-categories III. Notes

  • Wednesday October 07th: Lifting calculus. Notes

  • Wednesday October 14th: Lifting calculus II. Notes

  • Wednesday October 28th: Alternative models. Notes

  • Wednesday November 4th: Alternative models II. Notes

  • Wednesday November 11th: Alternative models end; joins. Notes

  • Wednesday November 18th: joins, slices and (co)limits. Notes

  • Wednesday November 25th: Colimits and Joyal extension Notes

  • Wednesday December 2nd: Grothendieck construction I Notes

  • Wednesday December 9th: Grothendieck construction II Notes Zoom link

Exercise sheets


J. Lurie, Higher Topos Theory

D. Cisinski, Higher categories and Homotopical Algebra

C. Rezk, Stuff on quasicategories

M. Groth, A short course on infinity-categories

E. Riehl, Categorical Homotopy Theory

J. Lurie, Kerodon