# Category: Higher category theory

Bicategory
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) assoc
En-ring
In mathematics, an -algebra in a symmetric monoidal infinity category C consists of the following data: * An object for any open subset U of Rn homeomorphic to an n-disk. * A multiplication map:for
Seifert–Van Kampen theorem
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fun
2-ring
In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a cate
Quasi-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a ge
Conglomerate (mathematics)
In mathematics, in the framework of one-universe foundation for category theory, the term "conglomerate" is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of
Extranatural transformation
In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
N-category
No description available.
Higher Topos Theory
Higher Topos Theory is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered f
N-group (category theory)
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, may be any natural number or infin
Tricategory
In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory. Whereas a weak 2-category is said to be a bicategory, a weak 3-category is said t
Higher category theory
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 st
N-category number
In mathematics, the category number of a mathematician is a humorous construct invented by Dan Freed, intended to measure the capacity of that mathematician to stomach the use of higher categories. It
Tetracategory
In category theory, a tetracategory is a weakened definition of a 4-category.
Stable ∞-category
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that * (i) It has a zero object. * (ii) Every morphism in it admits a and cofiber. * (iii) A triangle in it is
Strict 2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a catego
∞-topos
In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of she
Timeline of category theory and related mathematics
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: * Categories of abstract algebraic structures including representation theory and univers
Weak n-category
In category theory, a weak n-category is a generalization of the notion of strict n-category where composition and identities are not strictly associative and unital, but only associative and unital u
∞-groupoid
In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial
Category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups. In some of the literature, 2-groups a
Double groupoid
In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
Fundamental groupoid
In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it capt
N-monoid
In category theory, a (strict) n-monoid is an n-category with only one 0-cell. In particular, a 1-monoid is a monoid and a 2-monoid is a strict monoidal category.
Higher-dimensional algebra
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract
Stabilization hypothesis
In mathematics, specifically in category theory and algebraic topology, the Baez–Dolan stabilization hypothesis, proposed in, states that suspension of a weak n-category has no more essential effect a
Homotopy hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geo
2-functor
In mathematics, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-fu
String diagram
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When