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Category whose hom sets have algebraic structure
ordinary category theory. An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in
Enriched_category
Generalization of category theory
sense of enriched categories. The same for other enriched models like topologically enriched categories. Topologically enriched categories (sometimes
Higher_category_theory
Generalization of category
Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms
2-category
Categorical treatment of topological spaces
In category theory, a discipline in mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff
Topological category (enriched category theory)
Topological_category_(enriched_category_theory)
Generalization of a category
coherent nerve of the category of ∞-categories. Precisely, let K be the simplicially-enriched category where an object is a small ∞-category and the hom-simplicial-set
Quasi-category
General theory of mathematical structures
Mathematics portal Applied category theory Domain theory Enriched category theory Glossary of category theory Group theory Higher category theory Higher-dimensional
Category_theory
Category admitting tensor products
also used in the definition of an enriched category. Monoidal categories have numerous applications outside category theory proper. They are used to define
Monoidal_category
Category with direct sums and certain types of kernels and cokernels
the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all
Abelian_category
Category enriched over the category of simplicial sets
mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often also called
Simplicially enriched category
Simplicially_enriched_category
Mathematical object that generalizes the standard notions of sets and functions
formalized in the concept of an enriched category. A category is called complete if all small limits exist in it. The categories of sets, abelian groups and
Category_(mathematics)
In mathematics, invertible homomorphism
as the category of topological spaces or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules)
Isomorphism
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
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
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)
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)
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
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)
Topics referred to by the same term
mathematics, topological category may refer to: A concept in categorical topology; see topological functor A category enriched over the category of topological
Topological_category
Set of arguments where two or more functions have the same value
common throughout category theory for any binary equaliser. In the case of a preadditive category (a category enriched over the category of Abelian groups)
Equaliser_(mathematics)
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
Type of category in category theory
equations. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the
Additive_category
American mathematician and philosopher (1937–2023)
various kinds of enriched category. In this framework, the hom-set between two objects is replaced by an object in some other category. A primary example
William_Lawvere
Overview of and topical guide to category theory
semantics. Category Functor Natural transformation Homological algebra Diagram chasing Topos theory Enriched category theory Higher category theory Categorical
Outline_of_category_theory
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
Embedding of categories into functor categories
The Yoneda lemma is a fundamental result in category theory, a branch of mathematics. It is an abstract result on functors of the type morphisms into a
Yoneda_lemma
Topics referred to by the same term
Look up enrichment or enrich in Wiktionary, the free dictionary. Enrichment or enriched may refer to: Data enrichment, appending data with context from
Enrichment
Legal remedy taking away a benefit wrongfully obtained
is primarily governed by the "principle of unjust enrichment": A person who has been unjustly enriched at the expense of another is required to make restitution
Restitution and unjust enrichment
Restitution_and_unjust_enrichment
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
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
Category theory constructs
Gregory Maxwell (1982), "Chapter 4 Kan extensions", Basic concepts of enriched category theory (PDF), London Mathematical Society Lecture Note Series, vol
Kan_extension
Mathematical construction used in homotopy theory
homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets
Simplicial_set
Category-theoretic construction
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces
Coproduct
Generalized object in 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
Product_(category_theory)
functors preserve limits. For example, if a metric space is viewed as an enriched category (see generalized metric space), then the Cauchy completion of it coincides
Completions in category theory
Completions_in_category_theory
Theorem in category theory
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. It is a broad abstract generalization of many diagonal arguments
Lawvere's_fixed-point_theorem
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
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
Quotient space of a codomain of a linear map by the map's image
cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps
Cokernel
Construction in category theory
any category, although their existence depends on the category that is considered. They are a special case of the concept of a limit in category theory
Inverse_limit
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
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
Topics referred to by the same term
simplicial category may refer to: Simplex category, the category of finite ordinals and order-preserving functions Simplicially enriched category, a category enriched
Simplicial_category
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
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
Special objects used in (mathematical) category theory
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely
Initial_and_terminal_objects
Mathematical category
category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category
Topos
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
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
Mathematical space with a notion of distance
identity in an enriched category. Since R ∗ {\displaystyle R^{*}} is a poset, all diagrams that are required for an enriched category commute automatically
Metric_space
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification
Refinement_(category_theory)
Simplicial set constructed from the objects and morphisms of a small category
the homotopy coherent nerve functor associates to a simplicially enriched category a simpllicial set; i.e., N h c : sSetCat → sSet . {\displaystyle N^{hc}:{\textbf
Nerve_(category_theory)
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
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
Category where each homset contains at most one morphism
a poset if the category is small). A thin category is sometimes assumed skeletal. Equivalently, a thin category is a category enriched over the initial
Thin_category
mathematics known as category theory, a cosmos is a symmetric closed monoidal category that is complete and cocomplete. Enriched category theory is often considered
Cosmos_(category_theory)
Injective homomorphism
Y {\displaystyle X\hookrightarrow Y} . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative
Monomorphism
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)
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
Construction in category theory
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances
Cone_(category_theory)
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
Category
more detail, this means that a category C is pre-abelian if: C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently
Pre-abelian_category
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
Special case of colimit in category theory
objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms
Direct_limit
Relation of categories in category theory
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e
Isomorphism_of_categories
Category whose objects and morphisms are inside a bigger category
In mathematics, specifically category theory, a subcategory of a category C {\displaystyle {\mathcal {C}}} is a category S {\displaystyle {\mathcal {S}}}
Subcategory
Characterizing property of mathematical constructions
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions
Universal_property
Adjunction between a category of co/presheaf under the co/Yoneda embedding
adjunction) (named after John R. Isbell) is a fundamental construction of enriched category theory formally introduced by William Lawvere in 1986. That is a duality
Isbell_duality
Aspect of category theory in mathematics
categories weakly enriched in symmetric monoidal categories". Theory and Applications of Categories. 24 (20): 564–579. arXiv:0909.5270. Rig category at
Rig_category
Mathematical concept
In category theory, an end of a functor S : C o p × C → X {\displaystyle S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a
End_(category_theory)
between the same category. enriched category Given a monoidal category (C, ⊗, 1), a category enriched over C is, informally, a category whose Hom sets are
Glossary_of_category_theory
Collection of maps which give the same result
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and
Commutative_diagram
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)
Aspect of category theory
In category theory, a coequalizer (or coequaliser) is a generalization of the quotient of a set by an equivalence relation to objects in an arbitrary category
Coequalizer
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
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
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
Categorical generalization of a function space in set theory
specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all
Exponential_object
Concept in category theory
In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all
Forgetful_functor
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
In mathematics, collection of classes
In mathematics, in the framework of a one-universe foundation for category theory, the term conglomerate is applied to arbitrary sets as a contraposition
Conglomerate_(mathematics)
Functors which are surjective and injective on hom-sets
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both
Full_and_faithful_functors
Concept in mathematics
S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules): C = M o d S and D = M
Tensor–hom_adjunction
product. The same way we can define a monoidal category as a one-object 2-category, i.e. an enriched category over ( C a t , × ) {\displaystyle (\mathbf {Cat}
Premonoidal_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
Type of category in mathematics
in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in
Closed_monoidal_category
In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such
Lift_(mathematics)
Gray category. A strict 3-category is defined as a category enriched over 2Cat, the monoidal category of (small) strict 2-categories. A weak 3-category is
3-category
edu/category/2023/05/metric_spaces_as_enriched_categories_ii.html#more https://golem.ph.utexas.edu/category/2022/01/optimal_transport_and_enriched_2.html#more
Generalized_metric_space
Homological construction in category theory
In mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation
Derived_functor
Functor type
mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors
Representable_functor
Generalization of a small category
coherence conditions expressing the axioms of category theory. See . Enriched category Double category Moerdijk, Ieke; Mac Lane, Saunders (1992). Sheaves
Internal_category
Variant of the notion of the center of a monoid, group, or ring to a category
In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the
Center_(category_theory)
Flour with nutrients added
40% of flour being enriched by 1942.[citation needed] In February 1942, the U.S. Army announced that it would purchase only enriched flour. This resulted
Enriched_flour
Australian mathematician
of Enriched Category Theory. Let V {\displaystyle {\cal {V}}} be a monoidal category, and denote by V {\displaystyle {\cal {V}}} -Cat the category of
Max_Kelly
Bi-universal property in category theory
In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero
Zero_morphism
In mathematics, specifically in category theory, a functor F : C → D {\displaystyle F:C\to D} is essentially surjective if each object d {\displaystyle
Essentially surjective functor
Essentially_surjective_functor
Monoidal category
Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C
Tannakian_formalism
Concept in mathematical category theory
Dai Tamaki. The Grothendieck construction and gradings for enriched categories. arXiv: 0907.0061. http://pantodon.jp/index.rb?body=Grothendieck_construction#cite
Category_of_elements
Study of categorified structures
bicategories, variable categories (also known as indexed or parametrized categories), topoi, effective descent, and enriched and internal categories. In higher-dimensional
Higher-dimensional_algebra
Classification of fissile nuclear material
Material (SSNM) refers to uranium-235 contained in uranium enriched above 20 percent (highly-enriched uranium), as well as any concentration of uranium-233
Special_nuclear_material
ENRICHED CATEGORY
ENRICHED CATEGORY
Boy/Male
Hebrew
God enriches.
Boy/Male
Muslim
The enricher, The emancipator
Boy/Male
Hebrew
God enriches.
Boy/Male
Muslim/Islamic
Servant of the Enricher
Girl/Female
Bengali, Hindu, Indian
Discreet; Enrich; Impressive; Advantage
Boy/Male
Indian
The enricher
Boy/Male
Indian
Enriched
Boy/Male
Indian
The enricher, The emancipator
Boy/Male
Indian, Tamil
Enriched with Love; One who Loves All
Boy/Male
Hebrew
God enriches.
Boy/Male
Tamil
The enriched one, Prosperous
Boy/Male
British, English, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
The Enriched One
Boy/Male
Muslim
Servant of the enricher
Boy/Male
Hindu
The enriched one, Prosperous
Boy/Male
Muslim
The enricher
Boy/Male
Hebrew
God enriches.
Boy/Male
Arabic
Servant of the Enricher
Boy/Male
Muslim
Enriched
Boy/Male
Arabic, Muslim
Servant of the Enricher
ENRICHED CATEGORY
ENRICHED CATEGORY
Boy/Male
Arabic, Muslim
Pious; Devout
Surname or Lastname
English
English : variant of Gooch, itself a variant of Goff.
Boy/Male
Hindu
The eye of victory
Boy/Male
Hindu, Indian, Tamil
Honourable Man
Boy/Male
Sikh
Favour or fortune of gods Love, Reservoir of Love, Mysterious secrets of Love, Essence of Love (1)
Boy/Male
English
Gentle. Famous Bearer: Clement Moore, writer of 'Twas the Night Before Christmas'.
Girl/Female
Hindu
Nourished, Defended, Loved
Boy/Male
Hindu
Enthusiasm
Girl/Female
Greek American French
Moon goddess.
Boy/Male
Greek
The guardian of Capricornians.
ENRICHED CATEGORY
ENRICHED CATEGORY
ENRICHED CATEGORY
ENRICHED CATEGORY
ENRICHED CATEGORY
n.
To fatten; to enrich.
imp. & p. p.
of Entice
v. t.
To supply with knowledge; to instruct; to store; -- said of the mind.
p. pr. & vb. n.
of Enrich
n.
That which adorns, enriches, or beautifies; something added by way of embellishment; ornament.
a.
Placed in a niche.
v. t.
To fertilize or enrich, as land.
v. t.
To enrich.
v. t.
To supply with ornament; to adorn; as, to enrich a ceiling by frescoes.
n.
One who enriches.
a.
Capable of being enticed.
a.
Enriched with spice and condiments; hence, exciting; piquant.
v. t.
To enrich; to exalt; to benefit.
v. t.
To place in a niche.
v. t.
To enrich.
v. t.
To make rich with manure; to fertilize; -- said of the soil; as, to enrich land by irrigation.
v. t.
To make rich with any kind of wealth; to render opulent; to increase the possessions of; as, to enrich the understanding with knowledge.
a.
Embellished with flowers of rhetoric; enriched to excess with figures; excessively ornate; as, a florid style; florid eloquence.
imp. & p. p.
of Enrich
a.
Bent into a curve; -- said of a bend or other ordinary.