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Ordinary categories capture composition that is strictly associative and unital. But many mathematical structures—homotopy types, cobordisms, derived schemes—demand composition that holds only up to coherent isomorphism, and those isomorphisms themselves must satisfy higher coherence conditions. This tension between strictness and weakness is the engine that drives higher category theory. The field has produced a dozen major frameworks, each responding to limitations in its predecessors, and the result today is a pluralistic landscape where several models coexist, specialized for different problems.
The first indirect approach to higher-dimensional structure came through Enriched Category Theory, developed from the 1960s onward. Instead of defining a category whose hom-sets are sets, enrichment allows hom-objects to live in any monoidal category V—topological spaces, chain complexes, simplicial sets. Enriched categories do not directly model higher morphisms, but they provide a flexible infrastructure for encoding higher homotopical information: a category enriched over simplicial sets, for instance, carries a natural notion of homotopy between morphisms. Enriched category theory remains a living methodological tool, used today as the foundation for simplicially enriched categories and as a general language for categorical algebra.
A more explicit step toward higher dimensions came with Bicategories (also called weak 2-categories), introduced by Jean Bénabou in 1967. A bicategory has objects, 1-morphisms, and 2-morphisms between 1-morphisms, but composition of 1-morphisms is only associative and unital up to coherent 2-isomorphisms. This was the first framework to embrace weakness explicitly. Bicategories revealed that coherence—the systematic management of associativity and unit isomorphisms—is not a defect but a structural feature. The coherence theorem for bicategories showed that every bicategory is equivalent to a strict 2-category, but the proof itself relied on the weak formulation. Bicategories directly inspired later attempts to define weak n-categories, both operadic and Trimble-style, by providing the template for how coherence laws propagate upward.
Two early attempts to generalize to higher dimensions took a strict route. Strict n-Categories, defined by Charles Ehresmann in the 1960s, require composition to be strictly associative and unital at all levels. They are algebraically clean and easy to define, but they fail to capture many naturally occurring structures. For example, the fundamental n-groupoid of a topological space is only weakly equivalent to a strict n-groupoid in low dimensions; for n ≥ 3, strictification destroys homotopical information. N-Fold Categories, also introduced by Ehresmann, take a different approach: an n-fold category is a set with n commuting category structures. This framework avoids the strictness problem by not requiring interchange laws to be invertible, but it proved too rigid for homotopy theory and remains a specialized tool, mainly used in algebraic topology and higher-dimensional rewriting.
The 1990s saw a burst of activity aimed at defining weak n-categories—structures where composition is associative and unital only up to coherent higher equivalences. Two major proposals emerged. Weak n-Categories (Operadic) , developed by Michael Batanin and others, uses operads to encode the coherence data: an n-operad acts on the collection of morphisms, and the weak n-category is an algebra for that operad. This approach is highly systematic and general, but its technical complexity made it slow to gain traction outside a dedicated community. Weak n-Categories (Trimble-style) , initiated by Todd Trimble, builds weak n-categories by iterating the process of enrichment: a weak 1-category is a category; a weak 2-category is a category enriched over weak 1-categories; and so on, with coherence conditions handled by a sequence of operads. Trimble-style definitions are conceptually elegant and connect naturally to topological quantum field theory, but they require careful handling of the enrichment base at each step. Both operadic and Trimble-style frameworks remain active research programs, but they have been largely superseded in practice by the (∞,1)-categorical revolution.
By the late 1990s, many higher category theorists realized that for homotopy-theoretic applications, the most urgent need was not a general definition of weak n-category, but a workable theory of (∞,1)-categories—structures where all k-morphisms for k > 1 are invertible. This narrowing of focus transformed the field. Four main models emerged, each encoding the same homotopical information in a different way.
Simplicially Enriched Categories (also called simplicial categories) are categories enriched over simplicial sets. They are the most direct generalization of topological categories and were used by William Dwyer and Daniel Kan in the 1980s to study homotopy theory of diagrams. Composition is strictly associative, but the simplicial enrichment captures homotopy coherence through the higher-dimensional simplices in the hom-objects. This model is intuitive and connects directly to simplicial model categories, but its strict composition can be a liability when constructing certain functors.
Segal Categories, introduced by Graeme Segal in the 1990s, take a different approach: a Segal category is a simplicial space (a functor from Δ^op to spaces) satisfying a Segal condition that encodes composition up to homotopy. This model relaxes strict composition entirely, replacing it with a homotopy coherent composition law. Segal categories are flexible and generalize naturally to higher dimensions, but they lack a completeness condition that ensures the correct notion of equivalence.
Complete Segal Spaces, developed by Charles Rezk in 2001, add exactly that completeness condition: a complete Segal space is a Segal category where the space of objects is equivalent to the space of homotopy equivalences. This condition ensures that the model captures the homotopy theory of (∞,1)-categories faithfully. Complete Segal spaces are particularly well-suited for studying the homotopy theory of (∞,1)-categories themselves, as they form a model category with excellent formal properties.
Quasi-Categories (also called weak Kan complexes) are simplicial sets that satisfy a weak horn-filling condition for inner horns. Introduced by André Joyal in the early 2000s and massively developed by Jacob Lurie, quasi-categories have become the dominant model for (∞,1)-categories. Their appeal is threefold: they are simple to define (a single condition on a simplicial set), they are deeply connected to classical simplicial homotopy theory, and Lurie's monumental work Higher Topos Theory and Higher Algebra built an entire infrastructure of (∞,1)-categorical mathematics on top of them. Quasi-categories now serve as the default language for derived algebraic geometry, topological field theory, and much of modern homotopy theory. The other three models remain in use—simplicially enriched categories for model category contexts, complete Segal spaces for formal homotopy theory, Segal categories for certain higher-dimensional generalizations—but quasi-categories have the largest user base and the most extensive toolkit.
Once the (∞,1)-categorical foundation was secure, attention turned to higher dimensions. (∞,N)-Categories generalize (∞,1)-categories to allow non-invertible morphisms up to level N, with all higher morphisms invertible. Several models exist: one can iterate the complete Segal space construction to obtain (∞,n)-categories as n-fold complete Segal spaces, or use Θ_n-spaces, or adapt the quasi-category approach via complicial sets. These frameworks are essential for topological quantum field theory, where the cobordism hypothesis (proved by Lurie using (∞,n)-categories) relates fully extended TQFTs to dualizable objects in symmetric monoidal (∞,n)-categories.
Higher Algebra, as developed by Lurie, is not a separate model but a systematic extension of quasi-category theory to algebraic structures: monoidal (∞,1)-categories, algebras and modules over them, E_n-algebras, and so on. Higher Algebra provides the language for derived algebraic geometry (where schemes are replaced by functors from simplicial commutative rings to ∞-groupoids) and for the study of factorization homology. It is the most active area of application today, connecting higher category theory to geometry, topology, and representation theory.
Today, higher category theory is a mature but still evolving field. The leading frameworks—quasi-categories, complete Segal spaces, and simplicially enriched categories—agree on the homotopy theory of (∞,1)-categories: they are all equivalent as model categories, and the choice between them is largely a matter of convenience. They disagree on which model is best for specific tasks: quasi-categories excel for algebraic constructions, complete Segal spaces for formal homotopy theory, and simplicially enriched categories for model categorical contexts. For (∞,n)-categories, the field is more pluralistic, with n-fold complete Segal spaces and Θ_n-spaces competing. The operadic and Trimble-style weak n-category definitions continue to be studied for their conceptual elegance and connections to higher algebra, but they have not displaced the (∞,1)-centric approach for mainstream applications. The central open question is whether a fully satisfactory definition of weak ω-category (with no invertibility constraints) can be given that is both tractable and useful—a goal that has driven the field since its inception and remains a frontier.