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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
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)
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
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
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
Special kind of category with "dual objects"
closed. The simplex category can be used to construct an example of non-symmetric compact closed category. The simplex category is the category of non-zero
Compact_closed_category
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
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
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
Generalization of 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
Higher_category_theory
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
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
topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all [ n ] = { 0 , 1 , ⋯ n } {\displaystyle [n]=\{0
Simplicial_diagram
Generalization of category
nerve N h c ( C ) {\displaystyle N^{hc}(C)} of a 2-category C is a simplicial set where each n-simplex is determined by the following data: n objects x
2-category
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
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 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
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)
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)
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
precisely a simple object in the category of (say left) modules. simplex category The simplex category Δ is the category where an object is a set [n] =
Glossary_of_category_theory
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)
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
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)
Mathematical concept
[n]} of Δ {\displaystyle \Delta } to the standard n {\displaystyle n} -simplex inside R n + 1 {\displaystyle \mathbb {R} ^{n+1}} . Finally there is a
End_(category_theory)
Multi-dimensional generalization of triangle
0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron
Simplex
elementary topos (plural topos or topoi) is a category which has properties making it resemble the category of sets. Elementary toposes can be used as models
Elementary_topos
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
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)
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
Set of arguments where two or more functions have the same value
proved that any equaliser in any category is a monomorphism. If the converse holds in a given category, then that category is said to be regular (in the
Equaliser_(mathematics)
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
Mathematical category with weak equivalences, fibrations and cofibrations
simplicial sets, which are presheaves on the simplex category). If C is a model category, then so is the category Pro(C) of pro-objects in C. However, a model
Model_category
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
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
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)
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
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
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
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
Type of category in mathematics
category M would also get the induced model category structure. A prototypical example is the simplex category or its opposite. It was introduced by Christopher
Reedy_category
Simplicial object in the category of topological spaces
object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological
Simplicial_space
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
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
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
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
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)
Algorithm for linear programming
Dantzig's simplex algorithm (or simplex method) is an algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and
Simplex_algorithm
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
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)
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
Contravariant functor to Set
Set-valued presheaf on the simplex category C = Δ {\displaystyle C=\Delta } . A directed multigraph is a presheaf on the category with two objects and two
Presheaf_(category_theory)
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
Aspect of category theory
1-simplex. Coequalizers can be large: There are exactly two functors from the category 1 having one object and one identity arrow, to the category 2 with
Coequalizer
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
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
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
For example, by definition, a simplicial set is a presheaf on the simplex category Δ and a representable simplicial set is exactly of the form Δ n = Hom
Density theorem (category theory)
Density_theorem_(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
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
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
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 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
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
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
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
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
mathematics, especially category theory, Joyal's theta category Θ {\displaystyle \Theta } is an alternative to the simplex category Δ {\displaystyle \Delta
Joyal's_theta_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
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
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
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
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
Viral disease caused by herpes simplex viruses
Herpes simplex, often known simply as herpes, is a viral infection caused by the herpes simplex virus. Herpes infections are categorized by the area of
Herpes
Species of flowering plant
Ruellia simplex, the Mexican petunia, Mexican bluebell, Britton's wild petunia, or Purple showers, is a species of flowering plant in the family Acanthaceae
Ruellia_simplex
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
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
Hypothesis in mathematical category theory
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces
Homotopy_hypothesis
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)
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
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
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
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)
Category enriched over the category of simplicial sets
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
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
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
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
a} . A typical case is when I = Δ {\displaystyle I=\Delta } is the simplex category or rather its opposite; in the latter case, the functor f {\displaystyle
Limits and colimits in an ∞-category
Limits_and_colimits_in_an_∞-category
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
In category theory, an abstract branch of mathematics, and in its applications to logic and theoretical computer science, a list object is an abstract
List_object
Simplicial set constructed from the objects and morphisms of a small category
below for details. Let C be a small category. There is a 0-simplex of N(C) for each object of C. There is a 1-simplex for each morphism f : x → y in C.
Nerve_(category_theory)
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)
Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more
Pre-abelian_category
Connects set theory with category theory
set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors
Categorification
Category theory constructs
Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits
Kan_extension
Abstract homotopical model for topological spaces
objects in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, a category in which every
∞-groupoid
Endofunctor on the category V of finite-dimensional vector spaces
In algebra, a polynomial functor is an endofunctor on the category V {\displaystyle {\mathcal {V}}} of finite-dimensional vector spaces that depends polynomially
Polynomial_functor
Functor that preserves short exact sequences
exact, but in ways that can still be controlled. Let P and Q be abelian categories, and let F: P→Q be a covariant additive functor (so that, in particular
Exact_functor
Object in category theory
In category theory in mathematics, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely
Natural_numbers_object
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