Research pathwaysCategory TheoryLesson available
∞-categories
Optional research direction: (∞,1)-categories as quasi-categories and other models, ∞-groupoids and the homotopy hypothesis, limits, adjunctions and the Yoneda lemma in the ∞-setting, and applications in homotopy theory and algebraic geometry.
math.cat.infinity-categories
Builds on
Required prerequisites, all the way down: 54
You will learn to
- Define simplicial sets and quasi-categories, and recognise nerves of categories and Kan complexes as examples.
- Explain why composition in a quasi-category is defined only up to a contractible space of choices.
- Survey the models of (∞,1)-categories (quasi-categories, complete Segal spaces, simplicial categories) and how they are compared.
Definition
- A simplicial set is a functor Δᵒᵖ → Set.
- A quasi-category is a simplicial set in which every inner horn Λᵏ[n] → X with 0 < k < n extends to the simplex Δ[n].
- A Kan complex is a simplicial set in which every horn extends; Kan complexes model ∞-groupoids.
Theorem
- The nerve of a category is a quasi-category, and a simplicial set is isomorphic to the nerve of a category if and only if its inner horns have unique fillers.
- Joyal and Lurie developed category theory inside quasi-categories, including limits, adjunctions and a Yoneda lemma.
- The main models of (∞,1)-categories are related by Quillen equivalences of model categories (Bergner, Joyal–Tierney, Lurie).
Common misconceptions
- Thinking ∞-categories are just categories with more arrows; their composition and coherence are homotopical.
- Treating the homotopy hypothesis as a single theorem in every formulation; it is a guiding principle, established for particular definitions.
Leads to
Optional discoveries
Further reading
- Jacob Lurie, Higher Topos Theory (2009)
- Emily Riehl and Dominic Verity, Elements of ∞-Category Theory (2022)
- Denis-Charles Cisinski, Higher Categories and Homotopical Algebra (2019)