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A Infinity CategoryEdit

An ∞-category represents a broad generalization of the familiar notion of a category, extended to include higher levels of structure that encode equivalences not just between objects, but between morphisms, morphisms between morphisms, and so on. In this framework, composition is not required to be strictly associative or unital; instead, these properties hold up to coherent higher homotopies. The refinement is not merely technical; it provides a flexible language for fields where equivalences, rather than equalities, govern the mathematics. ∞-category and related ideas sit at the crossroads of category theory, homotopy theory, and modern algebraic geometry, and they underpin a great deal of work in derived mathematics, topological field theories, and beyond.

There are several compatible ways to formalize ∞-categories, each offering its own advantages depending on the problem at hand. The main approaches include quasi-categories (also called weak Kan complexes) quasi-category, complete Segal spaces complete Segal space, and various model-categorical frameworks built to model higher categorical data. In practice, researchers use these models interchangeably up to suitable notions of equivalence, which makes the subject robust to the choice of presentation. Within this landscape, A∞-categories provide a closely related, more algebraic perspective where composition is associative only up to a hierarchy of higher operations. A∞-categorys and ∞-categories interact fruitfully in areas such as derived categories, mirror symmetry, and higher topos theory. derived categorys and homological mirror symmetry are notable touchpoints where the ideas meet concrete computations.

Models and foundations

Quasi-categories

Quasi-categories are simplicial sets satisfying an inner horn-filling condition that encodes higher composition in a way that mirrors the composition of functors in ordinary category theory, but up to coherent homotopy. This model supports a flexible notion of functor (category theory) and natural transformations, while capturing the necessary higher-dimensional data for homotopy coherence. ∞-categorys realized as quasi-categories provide a convenient framework for doing homotopy-coherent higher algebra. quasi-category

Complete Segal spaces

Complete Segal spaces present ∞-categories as certain simplicial spaces with Segal conditions that enforce composition up to homotopy, together with a completeness condition that identifies equivalences with fully faithful, essentially surjective functors. This approach is closer to the language of homotopical algebra and offers a different set of technical tools for constructing and manipulating higher categories. complete Segal space

Model categories and Dwyer–Kan localization

Model categories give a setting to perform homotopy theory inside a categorical framework, and Dwyer–Kan localization supplies a mechanism to invert a chosen class of weak equivalences, producing a homotopy category enriched with higher data. These ideas provide a bridge between concrete algebraic models and the abstract ∞-categorical viewpoint. model category Dwyer–Kan localization

A∞-categories

A∞-categories generalize associative composition to a hierarchy of higher operations satisfying A∞-relations. They are particularly natural in contexts like derived categories of dg-categories and Fukaya categories, where algebraic structures carry non-strict associativity that is controlled by higher morphisms. A∞-categorys form a useful complement to the top-down ∞-categorical picture in many applications. A∞-algebra

Key constructions and notions

  • Objects and morphisms: At the base level, an ∞-category has objects and morphisms between them, but also higher morphisms between morphisms, and so forth, organized in a coherent way. category theory homotopy theory
  • Higher compositions: Composition is defined up to higher homotopies, with coherence conditions that ensure compatibility across all levels. This differs from the strict equalities found in ordinary categories. quasi-category
  • Equivalences: A central idea is that certain morphisms are equivalences (invertible up to higher homotopy) rather than strictly invertible. This mirrors the role of homotopy equivalence in topology. equivalence (category theory)
  • Functors and natural transformations: The notion of a functor extends to higher contexts, and natural transformations become higher-dimensional data. functor (category theory) natural transformation
  • Limits and colimits: ∞-categorical analogs of limits and colimits capture universal constructions in a homotopy-coherent way, with derived and homotopical interpretations. limit (category theory) colimit (category theory)
  • Stabilization: Stable ∞-categories generalize triangulated categories, providing a natural home for phenomena in stable homotopy theory and derived algebraic geometry. stable ∞-category triangulated category

Important families and examples

  • Derived categories and enhancements: Many derived categories arising in algebra and geometry admit ∞-categorical enhancements, making higher-homotopical methods available. derived category
  • Fukaya categories and mirror symmetry: In symplectic geometry, A∞-categories encode A∞-structures on Floer complexes, playing a central role in homological mirror symmetry. Fukaya category homological mirror symmetry
  • Derived algebraic geometry: The ∞-categorical framework is essential for formulating and working with derived stacks and related objects in modern algebraic geometry. derived algebraic geometry
  • Topological field theories: The language of ∞-categories informs the construction and study of extended topological quantum field theories, where higher morphisms encode cobordisms and their compositions. topological quantum field theory

Applications and impact

The ∞-categorical point of view unifies and clarifies many constructions across mathematics. In algebra and geometry, it provides a robust setting for derived methods, deformation theory, and moduli problems. In topology, it streamlines the handling of spectra, (co)homology theories, and higher invariants. In mathematical physics, higher categories furnish a natural algebraic backdrop for organizing field theories and their state spaces. The flexibility of higher coherence makes certain constructions more canonical and less sensitive to the choice of presentation, which can translate into clearer theorems and more transferable techniques. category theory homotopy theory topological quantum field theory

Controversies and debates

The field features ongoing discussions about foundational choices and practical trade-offs. Supporters of the ∞-categorical program emphasize the unifying power of a framework that handles all higher coherences in a single language, arguing that it reduces ad hoc adoptions of specialized tools and clarifies complex constructions. Critics sometimes point to the steep learning curve and the proliferation of models, arguing that the landscape can be technically demanding and that results should be communicated with greater concreteness and hands-on applicability. Proponents counter that presenting mathematics via higher coherence leads to more robust theorems and more natural interpretations of equivalence, despite the initial complexity. In practice, the community often adopts a model that best suits the problem at hand, while recognizing that true equivalence classes of ∞-categories behave well across different presentations. quasi-category complete Segal space model category A∞-category

Foundational questions also surface around set-theoretic size, universes, and the extent to which one should commit to a single foundational framework or maintain a palette of compatible approaches. Some researchers seek to keep the emphasis on computability and explicit constructions, while others prioritize broad abstract principles that can be deployed across diverse areas. Across these discussions, the pragmatic aim remains: to equip mathematicians with reliable, flexible tools for organizing complex, higher-dimensional relationships that arise in modern theory. ZFC universe (mathemat=set theory)

See also