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Generalization of category
In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat
2-category
Topics referred to by the same term
classes/categories Category of being Categories (Aristotle) Categoriae Decem Category (Kant) Categories (Peirce) Category (Vaisheshika) Stoic categories Category
Category
Topics referred to by the same term
Category 2 or Category II may refer to: Category 2 cable, a grade of unshielded twisted pair cabling Category 2 tropical cyclone, on any of the Tropical
Category_2
Category whose objects are small categories and whose morphisms are functors
morphisms are functors between categories. Cat may actually be regarded as a 2-category with natural transformations serving as 2-morphisms. The initial object
Category_of_small_categories
General theory of mathematical structures
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the
Category_theory
Collection of objects and morphisms
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked
Category_(mathematics)
Generalization of category theory
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)
Higher_category_theory
Unshielded twisted pair cabling for telephone and data communications
Category 2 cable, also known as Cat 2, is a grade of unshielded twisted pair cabling designed for telephone and data communications. The maximum frequency
Category_2_cable
Tropical cyclone intensity scale
location, which means a Category 2 hurricane that hits a major city will likely do far more cumulative damage than a Category 5 hurricane that hits a
Saffir–Simpson_scale
International standard for electrical and optical cables
100 kHz using Category 1 cable and connectors Class B: Up to 1 MHz using Category 2 cable and connectors Class C: Up to 16 MHz using Category 3 cable and
ISO/IEC_11801
equivalences. A (1, 1)-category is an ordinary category. A (2, 2)-category is a 2-category. A (2, 1)-category is a 2-category in which all 2-morphisms are invertible
(n,_m)-category
Categories of requirements for football stadiums set by UEFA
UEFA stadium categories are categories for football stadiums laid out in UEFA's Stadium Infrastructure Regulations. Using these regulations, stadiums
UEFA_stadium_categories
A Category 5 Pacific hurricane is a tropical cyclone in the Pacific Ocean that reaches Category 5 intensity on the Saffir–Simpson hurricane scale, the
List of Category 5 Pacific hurricanes
List_of_Category_5_Pacific_hurricanes
Category whose hom sets have algebraic structure
In category theory, a branch of mathematics, an enriched category generalizes the idea of a locally small category by replacing hom-sets with objects
Enriched_category
Generalization of a category
specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex
Quasi-category
Operation in algebra and mathematics
category theory, a branch of mathematics, a monad is a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category to
Monad_(category_theory)
Canadian classification for cable TV channels
television providers. It replaces the previous category A, category B, category C (instead split into the categories of "mainstream sports" and "national news")
Discretionary_service
'.Pro', '.1', and '.2'. For example, a race rated 1.1 equates to a one-day, category 1 race. A race classification ‘U’ (e.g. 2.2U) denotes an U-23 race
UCI_race_classifications
Category where each homset contains at most one morphism
Viewing a 2-category as an enriched category whose hom-objects are categories, the hom-objects of any extension of a thin category to a 2-category having
Thin_category
2-category. The pasting theorem for strict 2-category guarantees that every 2-categorical pasting scheme defines a unique composite 2-cell in every 2-category
Pasting_theorem
Category in which all small limits exist
In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C (where
Complete_category
Mathematical category with weak equivalences, fibrations and cofibrations
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences'
Model_category
Generalization of a category
of a 2-category leads to that of an n-category, iterating the process for a double category leads to that of an n-fold category. Double categories can
Double_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
Stable_∞-category
Generalized object in category theory
each of the given objects. Fix a category C . {\displaystyle C.} Let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be objects of C . {\displaystyle
Product_(category_theory)
Topics referred to by the same term
Category 4 or Category IV may refer to: Category 4 cable, a cable that consists of four unshielded twisted-pair wires Category 4 fireworks, British fireworks
Category_4
Category admitting tensor products
In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle
Monoidal_category
Aspect of category theory in mathematics
In category theory, a rig category (also known as bimonoidal category or 2-rig) is a category equipped with two monoidal structures, one distributing over
Rig_category
Category with direct sums and certain types of kernels and cokernels
prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are regular
Abelian_category
In mathematics, especially category theory, the homotopy category of an ∞-category C is the category where the objects are those in C but the hom-set
Homotopy category of an ∞-category
Homotopy_category_of_an_∞-category
Type of category in category theory
In mathematics, specifically in category theory, an additive category is a preadditive category admitting all finitary biproducts. There are two equivalent
Additive_category
Category whose objects are sets and whose morphisms are functions
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between
Category_of_sets
A Category 5 Atlantic hurricane is a tropical cyclone that reaches Category 5 intensity on the Saffir–Simpson hurricane wind scale, within the Atlantic
List of Category 5 Atlantic hurricanes
List_of_Category_5_Atlantic_hurricanes
especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors a strict 3-category, a semi-strict
3-category
Category whose objects are R-modules and whose morphisms are module homomorphisms
algebra, given a ring R {\displaystyle R} , the category of left modules over R {\displaystyle R} is the category whose objects are all left modules over R
Category_of_modules
Type of category in category theory
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified
Cartesian_closed_category
Classification system for prisoners
United Kingdom, prisoners are divided into four categories of security. Each adult is assigned to a category according to their crime, sentence, the risk
Prisoner security categories in the United Kingdom
Prisoner_security_categories_in_the_United_Kingdom
Category theory
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli
Kleisli_category
Type of quotient object in mathematics
quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally
Quotient_category
Category equipped with a faithful functor to the category of sets
mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor
Concrete_category
Mathematics construct
comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to
Comma_category
Lusternik-Schnirelmann category. Thus, in dimensions 2 and 3, the two invariants coincide. In dimension 4, the systolic category is known to be a lower
Systolic_category
Category whose objects are rings and whose morphisms are ring homomorphisms
many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper. The category Ring is a concrete category meaning
Category_of_rings
Mathematical category formed by reversing morphisms
In category theory, a branch of mathematics, the opposite category or dual category C op {\displaystyle C^{\text{op}}} of a given category C {\displaystyle
Opposite_category
above inclusion is given by a localization of an ∞-category. In Kerodon, the subcategory of a 2-category C obtained by removing non-invertible morphisms
Core_of_a_category
Topics referred to by the same term
Category 5 may refer to: Category 5 (album), an album from rock band, FireHouse Category 5 cable, used for carrying data Category 5 computer virus, as
Category_5
Generalization of a small category
categories in a fixed category into a 2-category. Let C {\displaystyle C} be a category with pullbacks. An internal category in C {\displaystyle C} consists
Internal_category
Ascribing an impossible property to a thing
A category mistake (or category error, categorical mistake, or mistake of category) is a semantic or ontological error in which things belonging to a particular
Category_mistake
Most general completion of a commutative square given two morphisms with same domain
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the
Pushout_(category_theory)
Concept in category theory
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise
Fibred_category
In category theory, a branch of mathematics, a C-indexed category is a pseudofunctor from Cop to Cat, where Cat is a 2-category of categories. Any indexed
Indexed_category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* (the internal
Rigid_category
Mathematical category whose hom sets form Abelian groups
specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian
Preadditive_category
Text from Aristotle's Organon
The Categories (Ancient Greek: Κατηγορίαι, romanized: Katēgoriai; Latin: Categoriae or Praedicamenta) is a text from Aristotle's Organon that enumerates
Categories_(Aristotle)
Concept in math
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the
Homotopy_category
Mathematical structures in category theory
In category theory, a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle
Functor_category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become
Localization_of_a_category
Risk of fetal injury due to the use of a pharmaceutical
The pregnancy category is an assessment of the risk of fetal injury due to a medication, if it is used as directed by the mother during pregnancy. It does
Pregnancy_category
Most general completion of a commutative square given two morphisms with same codomain
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit
Pullback_(category_theory)
Indexed collection of objects and morphisms in a category
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in
Diagram_(category_theory)
Generalization of the kernel of a homomorphism
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels
Kernel_(category_theory)
Product of two categories, in category theory
the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept
Product_category
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
Category of non-empty finite ordinals and order-preserving maps
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving
Simplex_category
Category whose objects are abelian groups and whose morphisms are group homomorphisms
In mathematics, the category A b {\displaystyle \mathbf {Ab} } has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype
Category_of_abelian_groups
Standardized data communications cable
ranges from 22 to 26 AWG. The standard for Category 6A (augmented Category 6) is ANSI/TIA-568.2-D (replaces 568-C.2), defined by TIA for enhanced performance
Category_6_cable
Mapping between categories
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic
Functor
Correspondence between properties of a category and its opposite
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite
Dual_(category_theory)
was introduced by Connes (1983). The cyclic category Λ has one object Λn for each natural number n = 0, 1, 2, ... The morphisms from Λm to Λn are represented
Cyclic_category
International classification for protected areas
IUCN protected area categories, or IUCN protected area management categories, are categories used to classify protected areas in a system developed by
IUCN protected area categories
IUCN_protected_area_categories
Category whose objects are groups and whose morphisms are group homomorphisms
In mathematics, the category G r p {\displaystyle \mathbf {Grp} } (or G p {\displaystyle \mathbf {Gp} } ) has the class of all groups for objects and group
Category_of_groups
Homological construction
In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense
Derived_category
In category theory, a premonoidal category is a generalisation of a monoidal category where the monoidal product need not be a bifunctor, but only to be
Premonoidal_category
Property of items within the grammar of a language
Grammatical Category. Anthropological Linguistics, Vol. 40, No. 2 (Summer, 1998), pp. 257-271 What is a grammatical category? - SIL.org "grammatical category" The
Grammatical_category
Mathematical concept
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products
Limit_(category_theory)
In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use
Filtered_category
Applications of category theory
Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer
Applied_category_theory
In ontology, the highest kinds or genera of entities
theory of categories concerns itself with the categories of being: the highest genera or kinds of entities. To investigate the categories of being, or
Theory_of_categories
References 2-category 1. A 2-category is a generalization of a category where there are also 2-morphisms between morphisms. 2. A (2, 1)-category is a 2-category
Glossary_of_category_theory
Within the North Atlantic Ocean, a Category 2 hurricane is a tropical cyclone that has 1-minute sustained wind speeds of between 83–95 knots (96–109 mph;
List of Category 2 Atlantic hurricanes
List_of_Category_2_Atlantic_hurricanes
Abstract mathematics relationship
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories
Equivalence_of_categories
Central object of study in category theory
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal
Natural_transformation
In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen
Exact_category
Category whose hom objects correspond (di-)naturally to objects in itself
In category theory, a branch of mathematics, a closed category is a special kind of category. In a locally small category, the external hom (x, y) maps
Closed_category
Audiovisual broadcasts by duration and content
This lists the longest-running television shows by category. The criterion for the longest-running show is the number of years the show has been on the
List of longest-running television shows by category
List_of_longest-running_television_shows_by_category
EC 2.3.2.2 Transglutaminase EC 2.3.2.13 Category:EC 2.4.2 Hypoxanthine-guanine phosphoribosyltransferase EC 2.4.2.8 Category:EC 2.5 Category:EC 2.5.1
List_of_enzymes
Simplicial set constructed from the objects and morphisms of a small category
In category theory, a discipline within mathematics, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms
Nerve_(category_theory)
Concept in mathematical category theory
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle
Symmetric_monoidal_category
Quiz game
Categories is a word game where players attempt to list words that fit into particular categories, all starting with the same letter. Players start by
Categories_(game)
Map (arrow) between two objects of a category
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures
Morphism
In mathematics, invertible homomorphism
Universal property Coherent isomorphism Balanced category 4 , 5 , 6 {\displaystyle 4,5,6} and 1 , 2 , 3 {\displaystyle 1,2,3} have both the usual order
Isomorphism
A Category 2 hurricane is a tropical cyclone which their peak intensity reaches the fourth-highest classification on the Saffir–Simpson hurricane wind
List of Category 2 Pacific hurricanes
List_of_Category_2_Pacific_hurricanes
Relationship between two functors abstracting many common constructions
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence
Adjoint_functors
Surjective homomorphism
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y
Epimorphism
Overview of and topical guide to category theory
category theory Category of sets Concrete category Category of small categories Category of vector spaces Category of graded vector spaces Category of
Outline_of_category_theory
Category theory concept
In mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering
Overcategory
Aspect of algebraic topology
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X {\displaystyle X} is the
Lusternik–Schnirelmann category
Lusternik–Schnirelmann_category
Mathematical object
The stable category of a Frobenius category is canonically a triangulated category. Dagger compact category Tannakian category Theorem 2.6 in Happel
Frobenius_category
In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows
Free_category
Category design is a business strategy and discipline that helps companies create, develop, and dominate new categories of products and services. Category
Category_design
travel, tourism, insurance
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