Infinity-categories

Course manual 2020/2021

Course content

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 functor 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).

Study materials

Literature

• 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

Objectives

• The student can explain clearly situations where usual category theory is not sufficient.
• The student can manipulate and simple computations with simplicial sets.
• The student can compute joins and slices of simplicial sets.
• The student can compute simple limits and colimits in infinity-categories.
• The student can determine when to use the straightening/unstraightening equivalence when generalising a definition from category theory.
• The student can determine which properties of quasicategories are model (in)dependent.

Teaching methods

• Lecture
• Self-study

Learning activities

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

The exam will be an oral examination, conducted via Zoom (or similar online tool).

The resit exam will have the same format as the exam.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• dr. S.M. Pepin Lehalleur

e-mail: simon.pepin.lehalleur@gmail.com