Higher Category Theory & ∞-Categories

Higher category theory extends classical category theory by allowing morphisms between morphisms at all levels, providing a powerful framework for homotopy theory, derived algebraic geometry, and topological quantum field theory.

Fundamental Concepts

n-Categories

An n-category consists of:

  • Objects (0-morphisms)
  • 1-morphisms between objects
  • 2-morphisms between 1-morphisms
  • ... continuing up to n-morphisms

With composition laws at each level satisfying coherence conditions.

The Homotopy Hypothesis (Grothendieck-Baez)

There is an equivalence of categories:

$$\text{n-Groupoids} \simeq \text{Homotopy n-Types}$$

where n-groupoids are n-categories with all morphisms invertible.

Interactive n-Category Explorer

∞-Category Structure

Homotopy Theory Applications

Homotopy Coherent Diagram

// Quasi-category definition in Lean def QuasiCategory (C : Type*) [Category C] : Type* := { X : SimplicalSet // ∀ (n : ℕ) (σ : Λ[n,k] → X), ∃! (τ : Δ[n] → X), τ ∘ ι = σ }

Groundbreaking Research

Higher Topos Theory
Jacob Lurie (2009)
arXiv:math/0608040

The foundational text establishing the theory of ∞-topoi and higher categorical logic.

On the Classification of Topological Field Theories
Jacob Lurie (2009)
arXiv:0905.0465

Establishes the cobordism hypothesis using higher category theory.

A Survey of (∞,1)-Categories
Julia Bergner (2010)
arXiv:math/0610239

Comprehensive survey of different models for (∞,1)-categories.

Homotopy Type Theory: Univalent Foundations
Univalent Foundations Program (2013)
HoTT Book

Revolutionary connection between type theory and homotopy theory.

Models of ∞-Categories

Quasi-Categories

Simplicial sets satisfying inner horn filling conditions.

Complete Segal Spaces

Simplicial spaces satisfying Segal and completeness conditions.

Simplicial Categories

Categories enriched over simplicial sets.

Applications Across Mathematics

Derived Algebraic Geometry

Higher categories provide the natural framework for derived schemes and stacks, allowing systematic treatment of "spaces with singularities".

Topological Quantum Field Theory

The cobordism hypothesis states that framed extended TQFTs are classified by fully dualizable objects in symmetric monoidal (∞,n)-categories.

Homotopy Type Theory

Provides computational foundations for mathematics where types are interpreted as spaces and identities as paths.

Leading Research Centers

  • Institute for Advanced Study - Jacob Lurie's group
  • MIT - Higher algebra and topology
  • University of Chicago - Homotopy theory group
  • Max Planck Institute - Mathematical logic and foundations
  • Carnegie Mellon - Homotopy type theory