Higher Categories

Job: organize morphisms between morphisms (and higher) so that homotopy-coherent algebra, derived geometry, and modern homotopy theory have a rigorous calculus of limits, functors, and universal properties.

Core

Ordinary category theory truncates at \(1\)-morphisms. Geometry and homotopy produce invertible higher cells (homotopies, quasi-isomorphisms, gauge transformations). \((\infty,1)\)-categories keep all invertible higher morphisms while treating \(n\ge 2\) morphisms as invertible — the setting of Lurie’s Higher Topos Theory / Higher Algebra.

Layer Content
Models quasicategories; complete Segal spaces; simplicial categories
Limits / colimits \(\infty\)-categorical universal properties; presentability
Algebra \(\infty\)-operads, \(\mathbb{E}_n\)-monoids — \(\infty\)-operads
Geometry derived / spectral schemes use \(\infty\)-categories of sheaves

Reading map

Page Focus
Foundations motivations; quasicategories vs complete Segal; limits; presentable \(\infty\)-cats
\(\infty\)-operads \(\infty\)-operads, \(\mathbb{E}_n\), little disks, applications

Dependency order

  1. Category theory (adjunctions, limits, Yoneda).
  2. Algebraic topology (simplicial sets, nerves, homotopy).
  3. Homological algebra (derived categories as homotopy categories of stable \(\infty\)-cats).
  4. Quasicategories \(\to\) presentable/stable \(\infty\)-cats \(\to\) operads / DAG.

Interfaces

Area Contact
Category theory \(1\)-categorical shadow: homotopy category \(h\mathcal{C}\)
Homological algebra \(D(\mathcal{A})\simeq h(\mathcal{D})\) for stable \(\infty\)-category \(\mathcal{D}\)
Derived AG \(\infty\)-categories of derived rings and stacks
Homotopy type theory synthetic account of \(\infty\)-groupoids
Topos theory \(\infty\)-topoi generalize Grothendieck topoi

Notation (paper-standard)

  • \(\mathcal{C},\mathcal{D}\) \(\infty\)-categories; \(\mathrm{Fun}(\mathcal{C},\mathcal{D})\); \(\mathcal{S}=\) spaces; \(\mathrm{Cat}_\infty\)
  • \(N(\mathbf{C})\) nerve; \(\mathfrak{C}[X]\) rigidification; Kan complex \(\Leftrightarrow\) \(\infty\)-groupoid
  • \(\mathrm{Pr}^L\), \(\mathrm{Pr}^R\); \(L\) presentable; \(\mathrm{Ind}\), accessible localizations

See it

Derived category is not enough. Mapping cones in \(D(\mathcal{A})\) are unique only up to non-canonical isomorphism; homotopy-coherent diagrams and derived \(\otimes\) need a stable \(\infty\)-category (or at least a DG-enhancement). Modern AG papers write \(R\mathrm{Hom}\), \(f_*\) as functors on \(\mathrm{QCoh}\) viewed as a stable presentable \(\infty\)-category — the \(1\)-categorical \(D_{\mathrm{qc}}(X)\) is the homotopy category.

Check

What is the homotopy category of an \(\infty\)-category \(\mathcal{C}\)?

Objects of \(\mathcal{C}\); morphisms \(\pi_0\,\mathrm{Map}_{\mathcal{C}}(X,Y)\); composition induced by composition of mapping spaces. Denoted \(h\mathcal{C}\). For quasicategories, \(h\mathcal{C}\) is the homotopy category of the simplicial set (Joyal).

Why are all \(n\)-morphisms for \(n\ge 2\) invertible in an \((\infty,1)\)-category?

By definition of the model: mapping spaces are \(\infty\)-groupoids (Kan complexes / spaces), so cells of dimension \(\ge 1\) in mapping spaces are invertible up to higher cells. Non-invertible higher morphisms require \((\infty,n)\)-categories.

Pitfalls

  • Model dependence: theorems should be invariant under the Quillen equivalences relating quasicategories, complete Segal spaces, etc.
  • Strict vs weak enrichment: simplicially enriched categories are a model; composition is strictly associative there but mapping spaces still encode homotopy.
  • “\(\infty\)-groupoid \(=\) space” is a theorem/design principle (homotopy hypothesis), not a slogan to misuse for non-invertible arrows.

Refs

  • Lurie, Higher Topos Theory (HTT); Higher Algebra (HA)
  • Joyal, The Theory of Quasi-Categories (notes)
  • Cisinski, Higher Categories and Homotopical Algebra
  • Groth–Schommer-Pries–Vidal surveys / Kerodon (online)

Next

Foundations · \(\infty\)-operads · Category theory · Derived AG · Homological algebra · Algebraic topology