en.unionpedia.org

Higher category theory, the Glossary

Index 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 structure behind those equalities.[1]

Table of Contents

  1. 41 relations: Abstract nonsense, Algebraic topology, André Joyal, Bicategory, Cartesian product, Categorification, Category (mathematics), Category of sets, Category of small categories, Category theory, Coherency (homotopy theory), Compactly generated space, Eilenberg–MacLane space, Enriched category, Functor, Fundamental groupoid, Hausdorff space, Higher-dimensional algebra, Homology (mathematics), Homotopy, Homotopy group, Homotopy theory, Invariant (mathematics), Jacob Lurie, Limit (category theory), Mathematics, Monoidal category, Morphism, Natural transformation, NLab, Nonabelian algebraic topology, Parametrization (geometry), Path (topology), Product (category theory), Quasi-category, Simplicial set, Singleton (mathematics), Strict 2-category, Topological space, Topology, Tricategory.

  2. Foundations of mathematics

Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them.

See Higher category theory and Abstract nonsense

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

See Higher category theory and Algebraic topology

André Joyal

André Joyal (born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory.

See Higher category theory and André Joyal

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) associative, but only associative up to an isomorphism.

See Higher category theory and Bicategory

Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets and, denoted, is the set of all ordered pairs where is in and is in.

See Higher category theory and Cartesian product

Categorification

In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues.

See Higher category theory and Categorification

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

See Higher category theory and Category (mathematics)

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. Higher category theory and category of sets are Foundations of mathematics.

See Higher category theory and Category of sets

Category of small categories

In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories.

See Higher category theory and Category of small categories

Category theory

Category theory is a general theory of mathematical structures and their relations. Higher category theory and Category theory are Foundations of mathematics.

See Higher category theory and Category theory

Coherency (homotopy theory)

In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".

See Higher category theory and Coherency (homotopy theory)

Compactly generated space

In topology, a topological space X is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below.

See Higher category theory and Compactly generated space

Eilenberg–MacLane space

In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name.

See Higher category theory and Eilenberg–MacLane space

Enriched category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.

See Higher category theory and Enriched category

Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

See Higher category theory and Functor

Fundamental groupoid

In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space.

See Higher category theory and Fundamental groupoid

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each that are disjoint from each other.

See Higher category theory and Hausdorff space

Higher-dimensional algebra

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures.

See Higher category theory and Higher-dimensional algebra

Homology (mathematics)

In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces.

See Higher category theory and Homology (mathematics)

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

See Higher category theory and Homotopy

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.

See Higher category theory and Homotopy group

Homotopy theory

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them.

See Higher category theory and Homotopy theory

Invariant (mathematics)

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects.

See Higher category theory and Invariant (mathematics)

Jacob Lurie

Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study.

See Higher category theory and Jacob Lurie

Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

See Higher category theory and Limit (category theory)

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Higher category theory and Mathematics

Monoidal category

In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.

See Higher category theory and Monoidal category

Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.

See Higher category theory and Morphism

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved.

See Higher category theory and Natural transformation

NLab

The nLab is a wiki for research-level notes, expositions and collaborative work, including original research, in mathematics, physics, and philosophy, with a focus on methods from type theory, category theory, and homotopy theory.

See Higher category theory and NLab

Nonabelian algebraic topology

In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras.

See Higher category theory and Nonabelian algebraic topology

Parametrization (geometry)

In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

See Higher category theory and Parametrization (geometry)

Path (topology)

In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis.

See Higher category theory and Path (topology)

Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.

See Higher category theory and Product (category theory)

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 generalization of the notion of a category.

See Higher category theory and Quasi-category

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way.

See Higher category theory and Simplicial set

Singleton (mathematics)

In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element.

See Higher category theory and Singleton (mathematics)

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.

See Higher category theory and Strict 2-category

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.

See Higher category theory and Topological space

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Higher category theory and Topology

Tricategory

In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.

See Higher category theory and Tricategory

See also

Foundations of mathematics

References

[1] https://en.wikipedia.org/wiki/Higher_category_theory

Also known as 3-category, 4-category, Higher categories, Higher dimensional category theory, Higher-dimensional category theory, N-category, N-category theory, Strict n-categories, Strict n-category, Strict ∞-categories.