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Higher category theory is the study of mathematical structures in which there are not only objects and morphisms (arrows) between them, but also morphisms between morphisms, and morphisms between those, and so on upward. Where ordinary category theory captures the idea of "processes" and "compositions of processes," higher category theory captures the idea of "processes between processes," "equivalences between equivalences," and the subtle ways in which such higher-level identifications can be organized. It is a subfield of category theory that has become central to modern algebraic topology, algebraic geometry, and mathematical physics, because many naturally occurring mathematical objects carry exactly this kind of higher-dimensional structure.
To understand the subject, begin with an ordinary category. It has objects, and for any two objects \(A\) and \(B\), a set of morphisms \(\mathrm{Hom}(A,B)\). Morphisms compose: given \(f: A \to B\) and \(g: B \to C\), there is a composite \(g \circ f: A \to C\). Composition is associative and unital, but only up to equality.
Now imagine a situation where morphisms themselves are objects of study. For instance, in topology, one might consider paths in a space as morphisms between points. A path from \(x\) to \(y\) and a path from \(y\) to \(z\) can be composed, but the composite is not literally equal to a given path from \(x\) to \(z\); it is only homotopic to it. A homotopy between paths is a "morphism between morphisms." But homotopies themselves can be homotoped, and so on. The resulting structure—points, paths, homotopies, higher homotopies—is a fundamental example of a higher category, specifically an \(\infty\)-groupoid.
The central difficulty of higher category theory is that when composition is only defined up to higher equivalence, one must specify not only the composition laws but also the coherence data: the homotopies that witness associativity, the homotopies between those homotopies, and so forth. There are many ways to organize this data, and the choice of organization leads to different models of higher categories. A large part of the field is devoted to understanding which models exist, how they relate, and which one is best suited for a given problem.
The origins of higher category theory lie in mid-twentieth-century algebraic topology. The need to organize homotopical information led topologists to consider structures such as simplicial sets, which are combinatorial models of spaces, and to notice that these carry higher compositional data. In the 1960s and 1970s, the notion of a "weak \(n\)-category" began to take shape, though it was not yet formalized. The work of John Boardman and Rainer Vogt in the early 1970s on "homotopy coherent diagrams" introduced what would later be recognized as a model of \((\infty,1)\)-categories, though they did not use that language.
A crucial conceptual step came with the recognition that there is a distinction between two kinds of higher morphisms. In an \((\infty,1)\)-category, all morphisms above level 1 are required to be invertible (up to higher equivalence). This is the structure that arises naturally in homotopy theory: the morphisms are maps, and the higher morphisms are homotopies, which are always invertible up to homotopy. In contrast, a general weak \(n\)-category allows non-invertible morphisms at every level. The former is far more tractable and has seen the most development; the latter remains a more difficult and less settled subject.
The modern era of higher category theory began in the 1990s and 2000s, driven by two major developments. First, André Joyal introduced the notion of a "quasi-category" (also called a weak Kan complex), a simplicial set satisfying a weak filling condition. Joyal showed that quasi-categories model \((\infty,1)\)-categories. Second, Jacob Lurie systematically developed the foundations of the subject in his book Higher Topos Theory (2009), showing that quasi-categories could serve as a workable foundation for large parts of mathematics, including derived algebraic geometry and the theory of \(\infty\)-topoi. Around the same time, other models were developed and compared: simplicial categories, Segal categories, complete Segal spaces, and others. The proof that these models are equivalent (in a suitable sense) was a major achievement, due to work of Joyal, Lurie, Clark Barwick, Daniel Kan, Julia Bergner, and others.
It is important to distinguish this modern development from earlier precursors. The homotopy coherent diagrams of Boardman–Vogt were not originally conceived as a foundation for higher category theory; they were a tool in homotopy theory. Similarly, the "weak \(n\)-categories" studied by various authors in the 1980s and 1990s (such as Carlos Simpson, Michael Batanin, and Tom Leinster) were attempts to define the general notion, but they did not immediately yield a usable theory. The modern field is defined by the successful construction of workable models and the demonstration that they can be used to do serious mathematics.
The most developed and widely used part of higher category theory is the theory of \((\infty,1)\)-categories. Several distinct models exist, each with its own advantages and disadvantages. They are not rival schools in the sense of incompatible philosophies; rather, they are different technical implementations of the same underlying concept, and the field has largely converged on the view that they are equivalent.
Quasi-categories. A quasi-category is a simplicial set in which every inner horn has a filler. Simplicial sets are combinatorial objects built from simplices of all dimensions; the filling condition ensures that composition is defined up to homotopy. This model is favored because it is simple to define and because the theory of simplicial sets is well developed. Its main drawback is that composition is not strictly associative; one must work with "up to homotopy" throughout, which can be technically demanding.
Simplicial categories. A simplicial category is a category enriched over simplicial sets: for any two objects, there is a simplicial set of morphisms, which encodes higher homotopical information. This model is conceptually close to ordinary category theory and is useful for constructing examples. However, it is not "weak" enough: composition is strictly associative, which is a stronger condition than one wants in general. To pass from a simplicial category to a quasi-category, one applies the homotopy coherent nerve; to go back, one uses a fibrant replacement. The two models are equivalent in a precise sense.
Complete Segal spaces. A Segal space is a simplicial space (a simplicial object in spaces) satisfying a Segal condition that encodes composition up to homotopy. A complete Segal space adds a condition ensuring that the notion of equivalence coincides with the notion of homotopy. This model is elegant and has good formal properties, but it is more abstract and less directly combinatorial than quasi-categories.
Relative categories and other models. There are also models based on categories with weak equivalences, such as relative categories and marked simplicial sets. These are useful for constructing examples from existing structures, but they are less commonly used as a primary framework.
The relationship between these models is now well understood. There is a diagram of Quillen equivalences between the model categories presenting these different notions, meaning that they all capture the same underlying \((\infty,1)\)-category theory. This is a genuine theorem, not a conjecture, and it is one of the foundational results of the field.
Beyond \((\infty,1)\)-categories, one can ask about \((\infty,n)\)-categories for \(n > 1\), where there are non-invertible morphisms at levels up to \(n\), and all higher morphisms are invertible. These arise naturally in many contexts, such as the study of cobordism categories in topological field theory, or the study of derived algebraic geometry. However, the theory is much less developed.
Several definitions of weak \(n\)-categories have been proposed, including those of Batanin (based on operads), Leinster, and Simpson. These definitions are quite different in flavor, and it is not fully settled whether they are all equivalent. For \((\infty,n)\)-categories, there are models such as \(\Theta_n\)-spaces and \(n\)-fold complete Segal spaces, which have been shown to be equivalent in some cases. But the general theory remains an active area of research, and many basic questions—such as how to define the correct notion of "equivalence" between such structures—are still being worked out.
A particularly important special case is the theory of \((\infty,2)\)-categories, which has seen significant progress in recent years. These are used in the study of derived algebraic geometry, where one wants to consider not only sheaves but also morphisms between sheaves, and in the study of higher representation theory.
The most consequential application of higher category theory has been the development of \(\infty\)-topos theory and higher algebra. An \(\infty\)-topos is a higher categorical analogue of a topos: a category of sheaves of spaces, satisfying certain exactness conditions. Lurie's Higher Topos Theory develops this theory in detail, showing that many results from ordinary topos theory generalize, and that \(\infty\)-topoi provide the right framework for derived algebraic geometry.
Higher algebra is the study of algebraic structures—such as rings, modules, and operads—internal to \((\infty,1)\)-categories. The key notion is that of an \(\infty\)-operad, which encodes operations that are associative and commutative only up to coherent homotopy. This allows one to define \(\mathbb{E}_n\)-algebras (algebras over the little \(n\)-disks operad), which interpolate between associative algebras (\(n=1\)) and commutative algebras (\(n=\infty\)). These structures are central to modern algebraic topology and mathematical physics, where they appear as the algebraic structures on spaces of observables in topological field theories.
The relationship between higher category theory and homotopy theory is particularly close. In fact, one can view \((\infty,1)\)-categories as a generalization of topological spaces: every space gives rise to an \(\infty\)-groupoid (the fundamental \(\infty\)-groupoid of the space), and conversely, every \(\infty\)-groupoid can be realized as a space. This equivalence, known as the homotopy hypothesis, is a theorem for the specific models used in practice, though it remains a guiding principle for the general theory.
The field today is characterized by a mature and well-developed theory of \((\infty,1)\)-categories, which is now used routinely by researchers in algebraic topology, algebraic geometry, and related fields. The quasi-categorical model, in particular, has become the default choice for many mathematicians, largely due to Lurie's extensive development of its foundations. There are also computer proof assistants, such as the UniMath project and the HoTT library, that formalize aspects of higher category theory, though this remains a niche activity.
At the same time, several major open problems remain. The theory of \((\infty,n)\)-categories for \(n \geq 2\) is still incomplete, and there is no consensus on the "correct" definition. The relationship between different models of weak \(n\)-categories is not fully understood. The theory of \(\infty\)-topoi is less developed than its ordinary counterpart, and many classical results have not yet been generalized. There is also ongoing work on the relationship between higher category theory and homotopy type theory, a new foundational system based on the interpretation of types as spaces; this has led to new insights into the structure of higher categories, but the full implications are not yet clear.
A notable feature of the current landscape is the coexistence of different technical frameworks. Some researchers prefer quasi-categories, others prefer complete Segal spaces, and still others work with simplicial categories. These are not competing paradigms in the sense of incompatible worldviews; they are different tools for the same job, and the choice is often a matter of convenience or taste. The field has largely moved past the "model wars" of the early 2000s, and the equivalence theorems have made it possible to translate results from one framework to another.
The deepest open question in the field is perhaps the search for a fully satisfactory theory of weak \(n\)-categories for all \(n\), including the correct notion of "equivalence" between such structures. This is not merely a technical issue: it is tied to fundamental questions about the nature of higher-dimensional algebra and the relationship between syntax and semantics in mathematics. Until this is resolved, higher category theory will remain a field with a solid foundation for its most important special case and an active frontier beyond it.