Research pathwaysCategory TheoryPlanned

∞-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

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Builds on

Higher categories Algebraic topology

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

Stable ∞-categories