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Higher category theory
Higher category theory
Higher category theory
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In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as the fundamental weak ∞-groupoid.

In higher category theory, the concept of higher categorical structures, such as (∞-categories), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features,[1] such as the Eilenberg-MacLane space.

Strict higher categories

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An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category theory. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n − 1)-morphisms gives an n-category.

Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an (n + 1)-category.

An n-category is defined by induction on n by:

  • A 0-category is a set,
  • An (n + 1)-category is a category enriched over the category n-Cat.

So a 1-category is just a (locally small) category.

The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.

While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories,[2] strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory; see the article Nonabelian algebraic topology, referenced in the book below.

Weak higher categories

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Main article: Weak n-category

In weak n-categories, the associativity and identity conditions are no longer strict (that is, they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in topology is the composition of paths, where the identity and association conditions hold only up to reparameterization, and hence up to homotopy, which is the 2-isomorphism for this 2-category. These n-isomorphisms must well behave between hom-sets and expressing this is the difficulty in the definition of weak n-categories. Weak 2-categories, also called bicategories, were the first to be defined explicitly. A particularity of these is that a bicategory with one object is exactly a monoidal category, so that bicategories can be said to be "monoidal categories with many objects." Weak 3-categories, also called tricategories, and higher-level generalizations are increasingly harder to define explicitly. Several definitions have been given, and telling when they are equivalent, and in what sense, has become a new object of study in category theory.

Quasi-categories

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Main article: Quasi-category

Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition. André Joyal showed that they are a good foundation for higher category theory by constructing the Joyal model structure on the category of simplicial sets, whose fibrant objects are exactly quasi-categories. Recently, in 2009, the theory has been systematized further by Jacob Lurie who simply calls them infinity categories, though the latter term is also a generic term for all models of (infinity, k) categories for any k.

Simplicially enriched categories

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Main article: Simplicially enriched category

Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look at them as a model for (infinity, 1)-categories, then many categorical notions (e.g., limits) do not agree with the corresponding notions in the sense of enriched categories. The same for other enriched models like topologically enriched categories.

Topologically enriched categories

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Main article: Topological category

Topologically enriched categories (sometimes simply called topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of compactly generated Hausdorff spaces.

Segal categories

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Main article: Segal category

These are models of higher categories introduced by Hirschowitz and Simpson in 1998,[3] partly inspired by results of Graeme Segal in 1974.

See also

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Notes

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Higher category theory

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Higher category theory is a branch of that generalizes ordinary by incorporating higher-dimensional morphisms, such as 2-morphisms between 1-morphisms and further levels of morphisms up to arbitrary dimensions, formalized in structures known as n-categories and ∞-categories. These higher morphisms are composed coherently up to , enabling the study of homotopical and homological phenomena in areas like and . ∞-categories, a central model, are typically presented as simplicial sets satisfying inner horn-filling conditions, capturing weak invertibility and higher coherences in a precise way. The development of higher category theory traces back to the mid-20th century, building on foundational work in by and in the 1940s, but gaining momentum with Daniel Quillen's introduction of model categories in 1967 to handle within a categorical framework. Key early contributions include J. Michael Boardman and Rainer Vogt's 1973 exploration of infinite-dimensional categories via weak Kan complexes, and André Joyal's refinement of quasi-categories in the 1980s and 1990s as a model for (∞,1)-categories. The field crystallized in the 2000s through Jacob Lurie's foundational texts, such as Higher Topos Theory (2009), which established ∞-categories and ∞-topoi as rigorous tools for higher-dimensional and . Central concepts in higher category theory include quasi-categories, which model ∞-categories via simplicial sets with horn-filling properties for coherent compositions; Cartesian and coCartesian fibrations, which encode functors and relative structures; and limits and colimits generalized to higher dimensions, often computed in presentable ∞-categories that admit all small colimits. Other notable ideas encompass stable ∞-categories for triangulated-like structures in derived geometry, monoidal ∞-categories for enriched settings like spectra in , and the homotopy hypothesis, positing that ∞-groupoids capture homotopy types of spaces. These tools unify disparate areas, with equivalences like Quillen adjunctions extended to ∞-settings to ensure homotopy invariance. Higher category theory finds applications in modeling derived categories and moduli spaces in algebraic geometry, computing generalized cohomology theories via Eilenberg-Mac Lane spectra, and advancing ∞-topos theory for descent and sheafification in higher geometry. It also underpins developments in homotopy type theory, where ∞-categories provide semantics for synthetic homotopy, and in operad theory for ∞-loop spaces. Ongoing research emphasizes accessible and presentable ∞-categories for computational aspects, with influences from works like Emily Riehl and Dominic Verity's Elements of ∞-Category Theory (2022).

Introduction

Definition and Basic Concepts

Higher category theory extends the foundational structures of ordinary category theory by incorporating multiple levels of morphisms, allowing for a richer framework to model relationships in mathematics and beyond. A higher category, specifically an n-category, is defined as a structure consisting of objects (0-morphisms), 1-morphisms between objects, 2-morphisms between 1-morphisms, and continuing up to n-morphisms between (n-1)-morphisms, with composition operations defined at each level that are associative and equipped with identities. This generalizes the familiar 1-category, which comprises merely a collection of objects and 1-morphisms (arrows) with a single composition operation satisfying associativity and the existence of identities. In an n-category, the composition of k-morphisms along a fixed (k-1)-morphism is associative but not necessarily strict, meaning that certain equalities hold only up to higher-dimensional isomorphisms rather than identically; this weak form of composition captures the essential homotopy-theoretic aspects of higher structures while avoiding overly rigid constraints. Each k-morphism has a source and target in the previous , forming a hierarchical where higher morphisms modify or relate compositions in lower dimensions. The of a higher category refers to the highest level of morphisms present: an n-category is finite-dimensional, with morphisms truncated beyond level n (higher cells are identities or absent), whereas an ω-category is infinite-dimensional, featuring k-morphisms for all finite k, enabling unbounded hierarchies suitable for modeling infinite processes or types. At its core, the underlying shape of a higher category is often formalized using a globular set, which consists of sets GkG_k for each k≥0k \geq 0, equipped with source and target maps sk,tk:Gk→Gk−1s_k, t_k: G_k \to G_{k-1} satisfying the globular condition sk−1∘sk=sk−1∘tks_{k-1} \circ s_k = s_{k-1} \circ t_k and tk−1∘sk=tk−1∘tkt_{k-1} \circ s_k = t_{k-1} \circ t_k, ensuring a tree-like compatibility across dimensions. Composition in higher categories is then specified via pasting diagrams, which are planar arrays of k-morphisms whose boundaries match, allowing vertical and horizontal compositions to be defined coherently without specifying explicit formulas for all cases.

Motivation and Importance

Higher category theory arises from the limitations of ordinary category theory in capturing the nuances of homotopy theory, where paths between maps necessitate 2-morphisms to encode higher homotopies. In the category of topological spaces, the homotopy category collapses homotopic maps into isomorphisms, but this process discards essential information about diagrams of spaces and higher-dimensional relationships; higher categories address this by incorporating 2-morphisms as homotopies between 1-morphisms, enabling a more faithful model of homotopy types through weak equivalences and Segal conditions that ensure higher homotopy coherences. A key importance of higher category theory lies in resolving coherence problems inherent in , particularly the rigidity of strict equalities, by introducing invertible higher morphisms that model equivalences up to . In structures like monoidal categories, strict associativity imposes excessive constraints, but invertible 2-morphisms and higher-dimensional coherences allow for flexible compositions where equivalences are defined via quasi-invertibility or inverses, providing a framework to handle homotopy-invariant notions without losing essential data. Higher category theory generalizes functors to pseudofunctors and natural transformations to modifications, facilitating weak preservation of composition and identities up to coherent in bicategories. A pseudofunctor between bicategories consists of functors on hom-categories equipped with invertible 2-cells for units and compositors, satisfying naturality and / coherence axioms, while modifications provide 2-cells between pseudonatural transformations that ensure vertical composition coherence; this extension is crucial for modeling higher-dimensional mappings in weak n-categories. From the perspective of higher-dimensional algebra, higher categories offer a unified framework for coherence theorems, extending Mac Lane's coherence theorem for monoidal categories—where every monoidal category is equivalent to a strict one via unique isomorphisms between tensor products—to higher dimensions through coherent 2-groups and weak n-categories. In this setting, coherent 2-groups, equipped with adjoint equivalences and higher associators, correspond bijectively to quadruples involving group cohomology, ensuring that compositions are unique up to specified higher isomorphisms, thus providing algebraic tools for topological and quantum field theoretic applications.

Historical Development

Early Foundations in Category Theory

Category theory emerged in the mid-20th century as a foundational framework for abstracting structural relationships in , particularly within . and introduced the concepts of categories, functors, and natural transformations in their seminal 1945 paper, motivated by the need to formalize equivalences between topological invariants such as homology theories. These structures provided a precise language for describing how mappings between mathematical objects preserve their essential properties, with categories consisting of objects and morphisms, functors mapping between categories while preserving composition, and natural transformations enabling comparisons between functors. This foundational work laid the groundwork for unifying disparate areas of under a common axiomatic umbrella. Building on these basics, subsequent developments enriched category theory with concepts that began to reveal the limitations of one-dimensional morphisms and suggest extensions to higher dimensions. In 1958, Daniel M. Kan defined , pairs of functors related by a natural of hom-sets that captured dualities appearing in homology computations and other topological constructions. further advanced the field in 1963 by introducing monoidal categories, which incorporate a operation compatible with the category's structure, as explored in his work on associativity and commutativity laws essential for algebraic applications. These innovations highlighted the need for structures accommodating multiple levels of abstraction, as natural transformations could be viewed as a rudimentary form of 2-morphisms between 1-morphisms (functors). The 1960s saw initial explorations into two-dimensional categorical structures, foreshadowing higher category theory. Jean Bénabou formalized bicategories in 1967, generalizing categories by allowing composition of 2-morphisms to be associative only up to specified isomorphisms, providing a weak framework for relations between categories that extended the strict associativity of ordinary categories. This work served as a precursor to broader higher-dimensional generalizations by addressing the coherence issues arising in multi-level compositions. Throughout the period from 1945 to 1970, profoundly influenced , particularly in the development of homology and theories. Eilenberg and Mac Lane's early contributions enabled axiomatic treatments of homology, as detailed in their collaborative works and Mac Lane's 1963 monograph on homology, which integrated categorical methods to compute topological invariants like chain complexes and exact sequences. Similarly, the axiomatization in Eilenberg and Steenrod's 1952 book on foundations relied on categorical principles to unify singular and approaches. These applications demonstrated 's power in abstracting and comparing functors across different spaces, solidifying its role as an indispensable tool in topological research during this era.

Emergence of Higher Structures

In the , the foundations for higher category theory began to emerge through the development of operads, which provided a systematic way to encode iterated operations and influenced subsequent work on higher-dimensional algebraic structures. J. Peter May introduced operads in his 1972 book The Geometry of Iterated Loop Spaces, formalizing them as a tool for studying loop spaces and . Concurrently, J. Michael Boardman and Rainer Vogt's 1973 monograph explored homotopy invariant algebraic structures using weak Kan complexes, providing an early model for infinite-dimensional weak categories. Building on earlier notions of 2-categories, G. M. Kelly and Ross Street published their "Review of the Elements of 2-Categories" in 1974, which rigorously defined strict 2-categories and their limits, extending strict notions to higher dimensions. During the 1980s and into the , attention shifted toward weak higher categories to capture more flexible structures analogous to bicategories, which had earlier provided a prototype for weakening associativity and unit laws up to . A pivotal advancement came in 1995 with Robert Gordon, A. J. Power, and Ross Street's Coherence for Tricategories, which proved a coherence theorem showing that every tricategory is triequivalent to a Gray category, where only interchange laws are weakened; this built on prior efforts and advanced the theory of weak 3-categories as a step toward general weak n-categories. In the late , John Baez's "Higher-Dimensional Algebra" series, particularly part III published in , popularized the field by defining weak n-categories via opetopes and exploring their connections to . The 2000s marked a surge in foundational models for infinite-dimensional weak categories, often called ∞-categories. André Joyal developed quasi-categories in the early 2000s as simplicial sets satisfying inner horn-filling conditions, offering a combinatorial model for (∞,1)-categories. Charles Rezk introduced complete Segal spaces in 2001, defining them as simplicial spaces that model homotopy-coherent categories through Segal conditions and completeness for univalence. Tom Leinster's 2004 book Higher Operads, Higher Categories synthesized these ideas, using generalized operads to unify strict and weak higher categories and providing a comprehensive reference. Culminating these efforts, Jacob Lurie's Higher Topos Theory in 2009 established a full framework for ∞-topoi using quasi-categories, enabling higher categorical generalizations of classical topos theory. Meanwhile, the n-Category Café blog, launched in 2006 by John Baez, David Corfield, and Urs Schreiber, became a key platform for disseminating ideas and fostering collaboration in the community.

Strict Higher Categories

Strict n-Categories

A strict n-category is defined inductively for finite n ≥ 0. A strict 0-category is simply a set, while a strict (n+1)-category is a category enriched over the category of strict n-categories, meaning that the hom-objects are strict n-categories and all structure maps, including associators and unitors, are identities rather than merely coherent isomorphisms. Equivalently, a strict n-category consists of a globular set—comprising collections of k-morphisms for 0 ≤ k ≤ n, with source and target maps satisfying globular shape conditions—equipped with vertical and horizontal composition operations that satisfy strict associativity, unitality, and interchange laws, where all higher-dimensional coherences are equalities. Representative examples illustrate this structure in low dimensions. A strict 1-category is an ordinary category, with objects as 0-morphisms and morphisms as 1-morphisms under strict composition. A is equivalent to a strict 2-category with a single 0-morphism (object), where the objects of the serve as 1-morphisms, the provides horizontal composition of 1-morphisms, and natural transformations act as 2-morphisms, all with strict associativity and unitality. Similarly, a braided can be viewed as a weak 3-category (tricategory) with one 0-cell and one 1-cell, where the braiding provides a modification inducing the necessary interchange coherences between horizontal and vertical compositions. Composition in strict n-categories proceeds via globular pasting schemes, which formalize the ways to compose arrays of k-morphisms along their boundaries for each dimension k ≤ n. These schemes ensure that horizontal and vertical compositions are strictly associative and interchangeable, meaning that the result of composing first horizontally and then vertically equals the result of composing first vertically and then horizontally, without any higher needed for coherence. This strict equality in all interchanges and associativities leads to the strictness problem: while many naturally arising higher categorical structures exhibit weak equalities (up to ), enforcing strictness severely limits the examples beyond low dimensions, as most interesting weak n-categories for n > 2 cannot be equivalent to strict ones. For instance, not every tricategory is triequivalent to a strict 3-category, though coherence results allow equivalence to semi-strict models like Gray-categories, where only certain associators are identities. Strict models, however, remain computationally simpler for formalizing and computing with finite-dimensional structures. Strict n-categories for finite n form the foundation for extending to infinite-dimensional cases, as detailed in the theory of strict ω-categories.

Strict ω-Categories

A strict ω-category is defined as a globular set equipped with partially defined binary composition operations at each k≥0k \geq 0, along with identity operations, such that all source-target compositions, whiskering compositions, and higher-dimensional interchanges are strictly associative and unital, obeying the exchange laws exactly as equalities. These structures generalize strict nn-categories to all finite dimensions without bound, where the kk-morphisms form categories over lower-dimensional ones in a strictly hierarchical manner. A canonical example of a strict ω-category is the category of strict ω-groupoids, where objects are globular sets with invertible cells at every level, and morphisms preserve the strict groupoidal structure, providing an archetypal model for homotopy types in a rigid algebraic setting. Another fundamental construction is the free strict ω-category on a globular set XX, obtained as the left adjoint to the forgetful functor from the category of strict ω-categories to the category of globular sets; this freely adjoins all necessary compositions and identities to XX while satisfying the strict laws. The Batanin-Leinster construction presents strict ω-categories as algebras over globular in the category of globular sets, where the underlying monad TT is cartesian and finitary, generated by pasting diagrams that encode all possible strict compositions coherently. This approach models all strict higher categories by ensuring that compositions arise from labelings of finite globular pasting schemes, providing a universal framework for generating the free structure on any globular set. (Note: Batanin's seminal work on monoidal globular operads extends this to weak variants, but the strict case aligns directly with .) Strict ω-categories exhibit significant rigidity, as all coherence conditions must hold via strict equalities rather than higher isomorphisms, rendering them rare in natural mathematical contexts where compositions typically commute only up to coherent homotopies; this limitation motivates the development of weak models that relax these equalities. A key structural result is that the category of strict ω-categories is enriched over itself, obtained by transfinitely iterating the enrichment starting from sets, which aligns with the Street nerve realizing these categories as complicial sets.

Weak Higher Categories

Bicategories and 2-Categories

A 2-category is a categorical structure consisting of 0-cells (objects), 1-cells (morphisms between 0-cells forming categories), and 2-cells (morphisms between parallel 1-cells), where both vertical composition of 2-cells and horizontal composition of 1-cells (whiskering with 2-cells) are strictly associative and unital, satisfying interchange laws and category axioms for each hom-category of 1-cells. This strictness distinguishes 2-categories from weaker variants, ensuring all compositions behave exactly as in ordinary categories without higher-dimensional adjustments. A prominent example is the 2-category Cat, whose 0-cells are small categories, 1-cells are functors, and 2-cells are natural transformations, with vertical composition given by vertical composition of natural transformations and horizontal composition by whiskering. In contrast, a bicategory relaxes these conditions to capture more flexible higher-dimensional structures, defined as a structure with 0-cells, 1-cells, and 2-cells where horizontal composition is associative and unital only up to specified invertible 2-cells called associators and left/right unitors, satisfying coherence conditions such as the pentagon identity for associators and for unitors. The 2-cells must be invertible for the unit and associator , but vertical composition remains strict, allowing bicategories to model situations where equality of composites is replaced by canonical . This weakening enables broader applications in areas like and , where strict equality is unnatural. The original definition of bicategories was introduced by Jean Bénabou in 1967. Key examples illustrate the utility of bicategories. The structure also forms a bicategory, where the weak equivalences arise naturally via invertible 2-cells (natural isomorphisms between functors), emphasizing how even strict 2-categories embed into the bicategorical framework without loss of coherence. Another example is the bicategory of spans in a category E\mathcal{E} with pullbacks, denoted Span(E)\mathbf{Span}(\mathcal{E}), where 0-cells are objects of E\mathcal{E}, 1-cells are spans (pairs of morphisms with common domain), composed via pullback, and 2-cells are commuting squares between spans; this construction generalizes relations and is central to . Similarly, the bicategory of profunctors, often denoted Prof\mathbf{Prof}, has small categories as 0-cells, profunctors (bifunctors Cop×D→Set\mathcal{C}^{\mathrm{op}} \times \mathcal{D} \to \mathbf{Set}) as 1-cells composed via coends, and natural transformations as 2-cells, providing a framework for generalized functors in enriched category theory. The coherence theorem for bicategories, established by Bénabou, asserts that any bicategory is biequivalent to a strict 2-category, meaning there exists a 2-category and pseudofunctors between them inducing equivalences on hom-categories, with all structure coherently related. This theorem extends Mac Lane's coherence for monoidal categories (a one-object bicategory), where the identity ensures a unique comparison up to natural equivalence for any of associators and unitors, guaranteeing that bicategories behave "as if strict" in diagrammatic reasoning without explicit tracking of higher .

Weak n-Categories and ∞-Categories

A weak nn-category is a higher categorical structure consisting of objects (0-morphisms), 1-morphisms between objects, 2-morphisms between 1-morphisms, and so on up to nn-morphisms, where the operations of composition and formation of units for kk-morphisms hold only up to coherently defined (k+1)(k+1)-isomorphisms for each k≤n−1k \leq n-1, and equivalences are defined as those weak morphisms that are invertible up to higher isomorphisms satisfying appropriate coherence conditions. This weakening of the strict associativity and unit axioms present in strict nn-categories allows for more flexible models that better capture the homotopical phenomena arising in and . The development of weak higher categories progressed beyond the 2-dimensional case of bicategories to tricategories, which are weak 3-categories in which compositions and units at each level are defined up to isomorphisms witnessed by higher-dimensional morphisms up to 3-morphisms, satisfying specified coherence conditions on those 3-morphisms, as formalized in the seminal work establishing their basic coherence theorem. This framework was then generalized to arbitrary finite dimensions by proposing foundational approaches to weak nn-categories using operadic and globular structures, emphasizing the need for coherence data at all levels to ensure well-defined compositions. An ∞\infty-category, also known as an (∞,1)(\infty,1)-category, is a weak ω\omega-category in which all kk-morphisms for k≥2k \geq 2 are equivalences, meaning they are invertible up to via higher-dimensional coherences, thereby modeling homotopy types where 1-morphisms represent maps up to and higher structure encodes homotopical inverses. A key property of (∞,1)(\infty,1)-categories is that limits and colimits, when they exist, are computed in a homotopy-invariant manner, preserving the weak equivalences and reflecting the homotopical nature of the structure. These concepts are illuminated by foundational hypotheses in higher category theory: Grothendieck's homotopy hypothesis posits an equivalence between the homotopy theory of topological spaces and the theory of ∞\infty-groupoids, where an ∞\infty-groupoid is an (∞,1)(\infty,1)-category in which all 1-morphisms are equivalences, effectively modeling spaces via their fundamental higher groupoids. Baez generalized this to propose that weak nn-groupoids should be equivalent to homotopy nn-types, and more broadly, that weak (n,r)(n,r)-categories provide algebraic models for truncated homotopy theories up to dimension nn with rr-morphisms behaving like equivalences.

Models for (∞,1)-Categories

Quasi-categories

A is a XX that admits fillers for all inner horns, meaning that for every n≥2n \geq 2 and 0<k<n0 < k < n, every map Λnk→X\Lambda^k_n \to X extends to a map Δn→X\Delta^n \to X. This condition, introduced by André Joyal, ensures that compositions of morphisms are defined up to coherent , modeling the weak invertibility of higher-dimensional morphisms in an (∞,1)(\infty,1)-category. Unlike Kan complexes, which fill all horns including outer ones, quasi-categories only require inner horn fillings, allowing for non-invertible 1-morphisms while treating higher morphisms as equivalences. Quasi-categories form the fibrant objects in Joyal's model structure on the category of simplicial sets, where weak equivalences are categorical equivalences (bijections on categories) and cofibrations are monomorphisms. In this structure, the mapping space Map⁡X(x,y)\operatorname{Map}_X(x,y) between objects x,y∈Xx,y \in X is a Kan complex, representing the space of morphisms up to . Joyal proved that this model structure is Quillen equivalent to other models for (∞,1)(\infty,1)-categories, such as simplicial categories, establishing quasi-categories as a robust framework for -coherent structures. The category of quasi-categories is Cartesian closed, supporting limits, colimits, and enriched structures essential for higher topos theory. Representative examples include the N(C)N(C) of an ordinary category CC, which is a quasi-category where 0-simplices are objects and 1-simplices are morphisms, with higher simplices encoding compositions. Kan complexes serve as quasi-categories modeling ∞\infty-groupoids, where all 1-morphisms are equivalences and the category is a . More generally, the coherent nerve of a simplicial category yields a capturing all coherent diagrams. The inner horn filling conditions imply that compositions in a quasi-category are unique up to , with the space of possible fillers being contractible, thus providing a coherent notion of weak composition. Jacob Lurie proved that quasi-categories present all (∞,1)(\infty,1)-categories via the homotopy coherent nerve functor from simplicial categories to quasi-categories, which forms a Quillen equivalence and thus an equivalence of homotopy theories. This result confirms that every (∞,1)(\infty,1)-category is equivalent to the homotopy coherent nerve of some simplicial category, making quasi-categories a complete model for the theory.

Complete Segal Spaces

A bisimplicial space XX, also known as a simplicial space, is a Segal space if it is Reedy fibrant and the Segal maps ϕn:Xn→X1×X0n\phi_n: X_n \to X_1 \times_{X_0}^n are weak equivalences for all n≥2n \geq 2. These maps encode the associativity of composition up to by ensuring that nn-tuples of morphisms are determined by their boundary data. A Segal space is complete if the s0:X0→X\hoequivs_0: X_0 \to X_{\hoequiv}, from the space of objects to the of homotopy equivalences, which is the subspace of 1-simplices consisting of components whose points are homotopy equivalences (i.e., invertible in the homotopy category), is itself a weak equivalence. Equivalently, completeness can be characterized by the Rezk of XX—a constructed to capture the of XX—admitting fillers for all inner horns, making it a . This model was introduced by Charles Rezk to provide a of (∞,1)(\infty,1)-categories in terms of simplicial spaces satisfying global Segal conditions. In a complete Segal space XX, the space of objects is given by the simplicial set X0X_0, while the space of morphisms between objects x,y∈X0x, y \in X_0 is the homotopy pullback X(x,y):=X1×X0×X0(x,y)X(x,y) := X_1 \times_{X_0 \times X_0} (x,y), which is itself an ∞\infty-groupoid modeling the homotopy type of mapping spaces. Composition of morphisms is defined via the Segal map ϕ2:X2→X1×X0X1\phi_2: X_2 \to X_1 \times_{X_0} X_1, which becomes an equivalence, allowing higher homotopies to witness associativity, units, and other categorical structure. The completeness condition ensures that every object is weakly equivalent to a "strict" one in the homotopy category, eliminating "redundant" components and guaranteeing that the homotopy category \Ho(X)\Ho(X) has objects precisely the π0(X0)\pi_0(X_0) with isomorphisms detected correctly. Rezk established that the ∞\infty-category of complete Segal spaces, equipped with a model structure where fibrant objects are precisely the complete Segal spaces, is equivalent to the ∞\infty-category of quasi-categories. Prominent examples of complete Segal spaces include the classifying diagram (or nerve) NCNC of an ordinary category CC, which is discrete in the simplicial direction and satisfies the Segal and completeness conditions, modeling CC as an (∞,1)(\infty,1)-category with discrete mapping spaces. More generally, the homotopy coherent nerve of a simplicial category—constructed as a bisimplicial space capturing higher-dimensional compositions—yields a complete Segal space up to weak equivalence, providing a model for (∞,1)(\infty,1)-categories enriched in simplicial sets. Complete Segal spaces offer advantages in modeling (∞,1)(\infty,1)-categories by presenting them as ∞\infty-groupoids of objects equipped with ∞\infty-groupoidal mapping spaces, where composition is enforced globally via the Segal maps, naturally aligning with the perspective that higher categories extend the of spaces under the Baez–Dolan homotopy hypothesis. This structure facilitates computations in by leveraging the bisimplicial framework to track both categorical dimensions and homotopical enrichments simultaneously.

Segal Categories

Segal categories provide a model for (∞,1)-categories through the lens of simplicial enrichment combined with a specified class of weak equivalences, emphasizing homotopy-invariant composition in mapping spaces. A Segal category consists of a small category C enriched over simplicial sets, equipped with a class of weak equivalences w ⊆ C, such that each mapping space Map_C(x, y) is a Kan complex and satisfies the Segal condition: for n ≥ 2, the natural map \Map∗C∗(x0,x1)×\Map∗C∗(x1,x1)⋯×\Map∗C∗(xn−1,xn−1)\Map∗C∗(xn−1,xn)→\Map∗C∗(x0,xn)\Map_*C*(x_0, x_1) \times_{\Map_*C*(x_1, x_1)} \cdots \times_{\Map_*C*(x_{n-1}, x_{n-1})} \Map_*C*(x_{n-1}, x_n) \to \Map_*C*(x_0, x_n) for a sequence of objects x0,…,xnx_0, \dots, x_n, induced by composition, is a weak equivalence in the Kan-Quillen model structure on simplicial sets. This ensures that composition is associative and unital up to coherent , with the weak equivalences w providing the necessary homotopy invariance. Key properties of Segal categories arise from their integration with model categorical techniques. The Dwyer-Kan simplicial localization functor L^{DK}_w C constructs, from any category C with weak equivalences w, a Segal category whose objects are those of C and whose mapping spaces encode homotopy classes of zigzags of morphisms in w, thereby presenting the homotopy category Ho(C) = \pi_0 L^{DK}_w C as the localization of C at w. This localization preserves the enriched structure, making Segal categories equivalent in the homotopy 2-category to quasi-categories via a Quillen equivalence between the model category of Segal categories and the Joyal model structure on quasi-categories. Julia E. Bergner's theorem establishes a left proper, cellular, simplicial model structure on the category of simplicial precategories (functors from Δ^{op} to simplicial sets with discrete object space), where the fibrant objects are precisely the Segal categories and the weak equivalences are the Dwyer-Kan equivalences (those inducing weak homotopy equivalences on mapping spaces after localization). Examples illustrate the flexibility of Segal categories in capturing homotopy-theoretic data. Any model category M gives rise to a Segal category via its hammock localization L^h M, a special case of Dwyer-Kan localization where mapping spaces are realized as geometric realizations of hammock diagrams (zigzags of cofibrations and acyclic fibrations), providing derived mapping spaces RHom_M(X, Y) that classify derived functors and homotopy limits/colimits. Relative categories, consisting of a category with a class of weak equivalences where hom-spaces are discrete (0-truncated simplicial sets), form a special case of Segal categories, as their Segal conditions reduce to ordinary categorical composition being the identity on components. In relation to other models, every Segal category is weakly equivalent (in the sense of Dwyer-Kan equivalences) to an ordinary simplicial category, where the Segal maps are isomorphisms rather than mere weak equivalences, but the explicit inclusion of w underscores the invariance essential for modeling (∞,1)-categories. The of a Segal category yields a Segal space, relating it briefly to the bisimplicial realization used in complete Segal spaces.

Other Models and Approaches

Simplicially Enriched Categories

A simplicial category, also known as a Δ\Delta-category or category enriched over , is a category C\mathcal{C} whose hom-objects C(x,y)\mathcal{C}(x,y) are simplicial sets for each pair of objects x,yx,y, equipped with composition maps that are simplicial set morphisms satisfying the usual associativity and unit axioms in the enriched sense. This structure models higher morphisms combinatorially, where nn-simplices in C(x,y)\mathcal{C}(x,y) represent nn-dimensional arrows between xx and yy, providing a flexible framework for encoding homotopy-coherent data without imposing strict equality on compositions. A key construction in this context is the homotopy coherent nerve N:CatΔ→sSetN: \mathrm{Cat}_\Delta \to \mathrm{sSet}, introduced by Jacob Lurie, which assigns to a simplicial category C\mathcal{C} the simplicial set N(C)N(\mathcal{C}) whose nn-simplices are homotopy-coherent diagrams in C\mathcal{C} of shape Δn\Delta^n. If the mapping spaces C(x,y)\mathcal{C}(x,y) are Kan complexes, then N(C)N(\mathcal{C}) is a , ensuring that the nerve captures the essential of C\mathcal{C}. The left adjoint to NN, known as the homotopy coherent realization functor, forms a Quillen adjunction with NN under the Kan-Quillen model structure on simplicial sets and a suitable model structure on simplicial categories, yielding a Quillen equivalence that identifies the homotopy theories of these models. Prominent examples include the category sSet\mathrm{sSet} of simplicial sets, which is simplicially enriched via the internal hom-functor (function complex) sSet(−,−)\mathrm{sSet}(-,-), where simplices encode transformations up to higher coherences. Another example arises in the context of ∞\infty-s: a category enriched over Kan complexes, such as the fundamental ∞\infty- of a space realized as a simplicial category with Kan complex hom-spaces, models the higher categorical structure where all morphisms are invertible up to . In higher category theory, simplicial categories serve as a general model for weak higher categories, bridging to (∞,1)(\infty,1)-categories through geometric realization of the , which produces a whose type reflects the higher structure of C\mathcal{C}. However, not every simplicial category directly yields a model for (∞,1)(\infty,1)-categories; additional conditions, such as the mapping spaces satisfying Segal-like maps for composition, are often required to ensure the nerve is a or complete Segal space. This enrichment provides a combinatorial foundation that avoids continuous while allowing integration with techniques for localization and equivalence detection.

Topologically Enriched Categories

A topologically enriched category consists of a category whose hom-objects are topological spaces and whose composition and identity assignment maps are continuous functions. More precisely, following Lurie's definition, it is a category over the of compactly generated weakly Hausdorff topological spaces, ensuring that the enrichment respects the topological structure in a way suitable for homotopy-theoretic applications. Topologically enriched categories provide a model for higher categories by incorporating continuous deformations, particularly for (∞,1)(\infty,1)-categories. A key feature is their ability to model ∞\infty-groupoids through Kan complexes in the category of topological simplicial sets, where a topological simplicial set is a simplicial object in the , and the Kan condition requires fillers for horns via continuous maps. The category of small topologically enriched categories admits a cofibrantly generated Quillen model structure, often referred to as the Thomason model structure in this context, where weak equivalences are functors inducing weak equivalences on nerves, cofibrations are cofibrations in a suitable sense, and fibrations are maps with the right lifting property. This model structure is Quillen equivalent to the Joyal model structure on quasi-categories, establishing topological categories as a valid model for (∞,1)(\infty,1)-categories with a focus on geometric . Examples include the category Top\mathbf{Top} of topological spaces, enriched over itself using the on mapping spaces YXY^X, where composition corresponds to continuous whiskering. Another is the fundamental ∞\infty- of a XX, modeled by the topological Kan complex obtained from the Sing(X)\mathrm{Sing}(X), regarded as a constant presheaf of topological spaces, capturing higher paths and homotopies in XX continuously. In geometric applications, topologically enriched categories are useful for modeling spaces equipped with higher-dimensional paths and their continuous deformations, facilitating the study of topological realizations of higher categorical structures, such as the geometric realization of the nerve of a topological category, which yields a space whose homotopy type reflects the higher categorical data. This aligns higher category theory with classical by providing a continuous framework for realizing abstract higher structures as concrete topological objects.

Applications

In Homotopy Theory

Higher category theory provides a framework for formalizing and extending classical , particularly through the lens of (∞,1)(\infty,1)-categories, which capture weak equivalences and homotopy coherences intrinsically. In this context, , introduced by Quillen, serve as presentations of (∞,1)(\infty,1)-categories, allowing the of spaces, spectra, and complexes to be studied in a unified manner. A key insight is that the derived (∞,1)(\infty,1)-category associated to a encodes the up to equivalence, without relying on explicit cofibrant or fibrant replacements. This is formalized by Lurie's , which establishes that for any combinatorial M\mathcal{M}, the Ho(M)\mathrm{Ho}(\mathcal{M}) is equivalent to the of the associated (∞,1)(\infty,1)-category Mderived\mathcal{M}^\mathrm{derived}, thereby embedding classical homotopy-theoretic constructions into the higher-categorical setting. Homotopy limits and colimits, central to algebraic topology, admit intrinsic definitions within (∞,1)(\infty,1)-categories modeled by quasi-categories. In a quasi-category C\mathcal{C}, a homotopy limit of a diagram F:D\op→CF: \mathcal{D}^\op \to \mathcal{C} is given by the limit in the homotopy category, but more precisely, it is realized as the homotopy pullback over the terminal object, generalizing the classical notion of homotopy pullbacks in simplicial sets or model categories. Similarly, homotopy colimits are defined dually as limits in the opposite quasi-category, enabling computations that avoid the need for functorial factorizations in model categories. These constructions extend Quillen's homotopy limits to arbitrary indexing categories and facilitate the study of derived functors in a coherent, up-to-homotopy manner. Stable (∞,1)(\infty,1)-categories further bridge higher category theory with , serving as enhancements of triangulated categories. The homotopy category of a stable (∞,1)(\infty,1)-category is triangulated, with the distinguished triangles arising from fiber sequences, which replace the exact triangles of triangulated categories while preserving their universal properties. This model applies directly to the (∞,1)(\infty,1)-category of spectra Sp\mathrm{Sp}, whose homotopy category computes stable homotopy groups, and to derived categories of abelian categories, unifying motivic and equivariant theories under a single framework. Lurie's development shows that natural triangulated categories in , such as those from or motives, are homotopy categories of stable (∞,1)(\infty,1)-categories, providing a coherent enhancement that resolves limitations of strict triangulations. The (∞,1)(\infty,1)-categorical viewpoint unifies Quillen model structures across diverse fields like , , and by treating derived functors as ordinary functors between (∞,1)(\infty,1)-categories, obviating the need for fibrant or cofibrant replacements in computations. This intrinsic derivation allows for seamless passage between model categories and their (∞,1)(\infty,1)-localizations, enabling the study of and Kan extensions in a homotopy-invariant way that applies uniformly to spaces, spectra, and sheaves.

In Algebraic Geometry and Physics

In derived algebraic geometry, higher category theory provides a framework for studying geometric objects that incorporate homotopical information, such as derived schemes and stacks. ∞-Stacks are modeled as sheaves of ∞-groupoids on a site, enabling the handling of higher coherences in descent data and equivalences, which is essential for defining derived moduli problems beyond classical schemes. Jacob Lurie's development of spectral algebraic geometry further extends this by replacing commutative rings with E_∞-ring spectra, allowing the structure sheaf to take values in the ∞-category of spectra and capturing derived intersections and cotangent complexes in a stable homotopy setting. This approach unifies algebraic geometry with stable homotopy theory, where affine spectral schemes correspond to E_∞-rings, and their quasi-coherent sheaves are modules over these spectra. In deformation theory, higher categories encode the higher-order structure of moduli stacks, particularly through derived stacks that resolve singularities and obstructions using simplicial commutative rings or dg-algebras. Derived stacks, as introduced by Bertrand Toën and Gabriele Vezzosi, allow the tangent complex to capture higher tangents and coherences, facilitating the study of deformations where classical stacks fail due to non-flatness or higher groups. For instance, the derived moduli stack of curves incorporates automorphisms and higher equivalences, providing a homotopically correct replacement for the coarse and enabling computations of deformation functors via simplicial resolutions. This framework resolves issues in classical deformation theory by treating obstructions in higher groups within an ∞-categorical setting. In physics, higher category theory models extended topological quantum field theories (TQFTs), where an n-dimensional extended TQFT is a symmetric monoidal functor from the ∞-category of framed bordisms to a target ∞-category with duals. The Baez-Dolan cobordism hypothesis posits that such TQFTs are determined by their value on the point, which is a fully dualizable object in the target category, classifying fully extended theories up to equivalence. Jacob Lurie outlined a proof of this hypothesis in the ∞-categorical setting (as of 2009), showing that the space of n-dimensional framed TQFTs is equivalent to the space of fully dualizable objects, with higher morphisms encoding the theory's structure; full proofs were later provided by others, such as Ayala and Francis for the framed case and Grady and Pavlov for geometric generalizations (up to 2021). This has implications for understanding anomalies and symmetries in quantum field theories through higher categorical data. A key example arises in and (CFT), where ∞-operads govern E_n-algebras that structure vertex operator algebras and correlation functions. In two-dimensional CFT, the of Riemann surfaces equips the theory with an E_2-operad structure, where sewing operations correspond to little disks compositions, capturing the associativity of operator product expansions. In , ∞-operads extend this to higher dimensions, modeling open-closed string interactions as algebras over the framed little disks operad, with higher coherences resolving ambiguities in and factorization homology. These structures link CFTs to topological strings, where E_n-algebras classify invariants like Gromov-Witten potentials in the derived setting.

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