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In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
Mapping between categories
identity functor is an endofunctor. Diagonal functor The diagonal functor is defined as the functor from D {\displaystyle D} to the functor category D
Functor
Relationship between two functors abstracting many common constructions
relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in
Adjoint_functors
Mathematical concept
index category. The functor category CJ may be thought of as the category of all diagrams of shape J in C. The diagonal functor Δ : C → C J {\displaystyle
Limit_(category_theory)
Construction in category theory
is nothing more than a functor category). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C. If
Cone_(category_theory)
Characterizing property of mathematical constructions
corresponding functor category. The diagonal functor Δ : C → C J {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal {C}}^{\mathcal {J}}} is the functor that maps
Universal_property
Central object of study in category theory
mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition
Natural_transformation
Generalized object in category theory
category C × C . {\displaystyle \mathbf {C} \times \mathbf {C} .} The diagonal functor Δ : C → C × C {\displaystyle \Delta :\mathbf {C} \to \mathbf {C} \times
Product_(category_theory)
Indexed collection of objects and morphisms in a category
type J one has a functor colim : CJ → C which sends each diagram to its colimit. The universal functor of a diagram is the diagonal functor; its right adjoint
Diagram_(category_theory)
Functors which are surjective and injective on hom-sets
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 properties
Full_and_faithful_functors
Embedding of categories into functor categories
category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category of representable functors and their
Yoneda_lemma
Category-theoretic construction
: C → C × C {\displaystyle \Delta :C\rightarrow C\times C} be the diagonal functor which assigns to each object X {\displaystyle X} the ordered pair (
Coproduct
Mathematical structures 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 F:C\to
Functor_category
Concept in category theory
specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure
Forgetful_functor
Mathematical set of all subsets of a set
contravariant power set functor, P: Set → Set and P: Set op → Set. The covariant functor is defined more simply as the functor which sends a set S to P(S)
Power_set
Mathematical category whose hom sets form Abelian groups
{\displaystyle C} and D {\displaystyle D} are preadditive categories, then a functor F : C → D {\displaystyle F:C\rightarrow D} is additive if it too is enriched
Preadditive_category
Concept in category theory
theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two
Monoidal_functor
Homological construction in category theory
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
category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an
Representable_functor
Construction for simplicial sets
twisted diagonal of a category is the category of elements of the Hom functor, the twisted diagonal of an ∞-category can be used to define the Hom functor of
Twisted diagonal (simplicial sets)
Twisted_diagonal_(simplicial_sets)
Type of category in mathematics
moreover exact. A logical functor is a functor between topoi that preserves finite limits and power objects. Logical functors preserve the structures that
Elementary_topos
Theorem in category theory
a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox
Lawvere's_fixed-point_theorem
Functor that preserves short exact sequences
particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations
Exact_functor
Family of type systems based on substructural logic
category theory point of view, no-cloning is a statement that there is no diagonal functor which could duplicate states; similarly, from the combinatory logic
Substructural_type_system
injection morphism to the l {\displaystyle l} -th component. Diagonal functor Diagonal embedding wikibooks:Category Theory/(Co-)cones and (co-)limits
Diagonal_morphism
Concept in mathematics
statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint
Tensor–hom_adjunction
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
Category whose hom sets have algebraic structure
properties. An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between
Enriched_category
other variants) in abstract algebra. diagonal functor 1. Given categories I, C, the diagonal functor is the functor Δ : C → F c t ( I , C ) , A ↦ Δ A {\displaystyle
Glossary_of_category_theory
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
Mathematical construction used in homotopy theory
topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were
Simplicial_set
Operation in algebra and mathematics
a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category to itself and two natural transformations η , μ {\displaystyle
Monad_(category_theory)
General concept and operation in mathematics
Δ between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c
Duality_(mathematics)
Generalization of category
(small) categories, where a 2-morphism is a natural transformation between functors. The concept of a strict 2-category was first introduced by Charles Ehresmann
2-category
Finest topology making some functions continuous
} be the diagonal functor from Top to the functor category TopJ (this functor sends each space X {\displaystyle X} to the constant functor to X {\displaystyle
Final_topology
General theory of mathematical structures
contravariant functor acts as a covariant functor from the opposite category Cop to D. A natural transformation is a relation between two functors. Functors often
Category_theory
Overview of and topical guide to category theory
Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Yoneda lemma Product (category
Outline_of_category_theory
Category whose objects are rings and whose morphisms are ring homomorphisms
there are forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and addition, respectively. Both of these functors have left adjoints
Category_of_rings
Construction in category theory
then just a contravariant functor I → C. Let C I o p {\displaystyle C^{I^{\mathrm {op} }}} be the category of these functors (with natural transformations
Inverse_limit
Type of category in category theory
must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints. When considering functors between R-linear
Additive_category
Simplicial object in the category of simplicial sets
\Delta \rightarrow \Delta \times \Delta } be the diagonal functor, then there is an induced functor δ ∗ = diag : b i s S e t → s S e t {\displaystyle
Bisimplicial_set
Concepts in algebraic topology
category of such diagrams is denoted SpacesI. There is a natural functor called the diagonal, Δ 0 : S p a c e s → S p a c e s I {\displaystyle \Delta _{0}:Spaces\to
Homotopy_colimit_and_limit
Generalization of a category
general simplicial set there is a functor τ {\displaystyle \tau } from sSet to Cat, the left-adjoint of the nerve functor, and for a quasi-category C, we
Quasi-category
Collection of maps which give the same result
diagram in a category C can be interpreted as a functor from an index category J to C; one calls the functor a diagram. More formally, a commutative diagram
Commutative_diagram
Type of category in mathematics
the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf
Topos
Type of category in category theory
The third condition is equivalent to the requirement that the functor –×Y (i.e. the functor from C to C that maps objects X to X×Y and morphisms φ to φ × idY)
Cartesian_closed_category
In category theory, a branch of mathematics, a conservative functor is a functor F : C → D {\displaystyle F:C\to D} such that for any morphism f {\displaystyle
Conservative_functor
Category with direct sums and certain types of kernels and cokernels
category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These
Abelian_category
Category whose only morphisms are the identity morphisms
discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects. Alternately, for a
Discrete_category
Collection of objects and morphisms
two categories compatible with their respective structures is called a functor. Well-known categories are denoted by a short capitalized word or abbreviation
Category_(mathematics)
mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense
Smooth_functor
Category whose objects and morphisms are inside a bigger category
There is an obvious faithful functor I : S → C {\displaystyle I:{\mathcal {S}}\to {\mathcal {C}}} , called the inclusion functor which takes objects and morphisms
Subcategory
Special objects used in (mathematical) category theory
categorical sum. It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will
Initial_and_terminal_objects
coaugmented functor. A coaugmented functor is a pair (L,l) where L:C → C is an endofunctor and l:Id → L is a natural transformation from the identity functor to
Localization_of_a_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)
Category mapping
category to the category Cat of (small) categories that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ
Pseudo-functor
In mathematics, invertible homomorphism
{\displaystyle FG=1_{D}} (the identity functor on D) and G F = 1 C {\displaystyle GF=1_{C}} (the identity functor on C). In a concrete category (roughly
Isomorphism
Product of two categories, in category theory
I} satisfy: given a family of functors f i : D → C i {\displaystyle f_{i}:D\to C_{i}} , there exists a unique functor f : D → P {\displaystyle f:D\to
Product_category
Concept in mathematical category theory
opposite of it is known as the twisted diagonal of C. Let X : I → Set {\displaystyle X:I\to {\textbf {Set}}} be a functor (thought of as a diagram) and E X
Category_of_elements
Category theory
notation mentioned in the “Formal definition” section above, define a functor F: C → CT by F X = X T {\displaystyle FX=X_{T}\;} F ( f : X → Y ) = ( η
Kleisli_category
Most general completion of a commutative square given two morphisms with same codomain
R, is given by the tensor product over R, and Spec is a contravariant functor, the pullback of two affine schemes Spec(A) and Spec(B) over Spec(R), usually
Pullback_(category_theory)
Category of non-empty finite ordinals and order-preserving maps
object is a presheaf on Δ {\displaystyle \Delta } , that is a contravariant functor from Δ {\displaystyle \Delta } to another category. For instance, simplicial
Simplex_category
Category theory constructs
Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician, Saunders Mac Lane titled
Kan_extension
unique functor F' : C(G) → D such that U(F')∘I=F, i.e. the following diagram commutes: The functor C is left adjoint to the forgetful functor U. Mathematics
Free_category
Category
pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive
Pre-abelian_category
Relation of categories in category theory
isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C. This
Isomorphism_of_categories
space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. [
Fundamental_groupoid
Monoidal category
gist of the theory is that the fiber functor Φ of the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional
Tannakian_formalism
Construction for categories
In category theory in mathematics, the twisted diagonal of a category (also called the twisted arrow category), which makes the morphisms of a category
Twisted diagonal (category theory)
Twisted_diagonal_(category_theory)
Branch of mathematics
Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development
Homological_algebra
Special case of colimit in category theory
the same as a covariant functor I → C {\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}} . The colimit of this functor is the same as the direct
Direct_limit
Category whose objects are sets and whose morphisms are binary relations
to make Rel a dagger category. The category has two functors into itself given by the hom functor: A binary relation R ⊆ A × B and its transpose RT ⊆
Category_of_relations
Abstract mathematics relationship
equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation
Equivalence_of_categories
Db(Y×Z), the composed functor ΦK2 ∘ {\displaystyle \circ } ΦK1 is also a Fourier-Mukai transform. The structure sheaf of the diagonal O Δ ∈ D b ( X × X )
Fourier–Mukai_transform
Concept in category theory
pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar
Fibred_category
Category equipped with a faithful functor to the category of sets
category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of the objects
Concrete_category
Category admitting tensor products
category where the functor X ↦ X ⊗ A {\displaystyle X\mapsto X\otimes A} has a right adjoint, which is called the "internal Hom-functor" X ↦ H o m C ( A
Monoidal_category
Map (arrow) between two objects of a category
diffeomorphisms. In the category of small categories, the morphisms are functors. In a functor category, the morphisms are natural transformations. For more examples
Morphism
Category theory concept
Forgetful functor Elementary topos Grothendieck topos Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative
Overcategory
Mathematical category formed by reversing morphisms
Dual (category theory) Duality (mathematics) Adjoint functor Contravariant functor Opposite functor "Is there an introduction to probability theory from
Opposite_category
Mathematics construct
1963 p. 13). The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object
Comma_category
Theory for associative algebras over rings
over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for
Hochschild_homology
{\displaystyle f:A\rightarrow B} in E {\displaystyle \mathbf {E} } then there is a functor f ∗ : E / B → E / A {\displaystyle f^{*}:\mathbf {E} /B\rightarrow \mathbf
Fundamental theorem of topos theory
Fundamental_theorem_of_topos_theory
Algebraic structure used in topology
derived functors of a left exact functor on an abelian category, while "homology" is used for the left derived functors of a right exact functor. For example
Cohomology
saying lim → − {\displaystyle \varinjlim -} is the left adjoint to the diagonal functor Δ − . {\displaystyle \Delta _{-}.} For this end, let α : f → Δ G {\displaystyle
Density theorem (category theory)
Density_theorem_(category_theory)
Type of quotient object in mathematics
equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor). Every functor F : C → D {\displaystyle
Quotient_category
Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext functor. A
Lift_(mathematics)
Forgetful functor Elementary topos Grothendieck topos Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative
N-group_(category_theory)
Category whose objects are abelian groups and whose morphisms are group homomorphisms
underlying function. This functor is faithful, and therefore A b {\displaystyle \mathbf {Ab} } is a concrete category. The forgetful functor has a left adjoint
Category_of_abelian_groups
Category whose hom objects correspond (di-)naturally to objects in itself
This is the internal hom [x, y]. Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to
Closed_category
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
property is that for a scheme T {\displaystyle T} , it represents the functor whose T {\displaystyle T} -valued points are the closed subschemes of P
Hilbert_scheme
Aspect of category theory
coequalizer as defined above, but with the added property that given any functor F : C → D, F(Q) together with F(q) is the coequalizer of F(f) and F(g)
Coequalizer
Higher categorical generalization of a topos
there is a small ∞-category C and an (accessible) left exact localization functor from the ∞-category of presheaves of spaces on C to X. A theorem of Lurie
∞-topos
Mathematical operation with two operands
Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's
Binary_operation
Concept in mathematical category theory
\circledast } ) is a closed symmetric monoidal category with the internal hom-functor ⊘ {\displaystyle \oslash } . The classifying space (geometric realization
Symmetric_monoidal_category
Connects set theory with category theory
replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was
Categorification
Categorical generalization of a function space in set theory
Z , Y {\displaystyle Z,Y} in C {\displaystyle \mathbf {C} } , then the functor ( − ) Y : C → C {\displaystyle (-)^{Y}\colon \mathbf {C} \to \mathbf {C}
Exponential_object
Generalization of category theory
the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms
Higher_category_theory
Array of numbers
particular, let ∅ {\displaystyle \varnothing } be an initial object. The functor ⊗ {\displaystyle \otimes } is distributive over coproducts; i.e., for all
Matrix_(mathematics)
Category whose objects are sets and whose morphisms are functions
category, the contravariant functors from C to Set are often an important object of study. If A is an object of C, then the functor from C to Set that sends
Category_of_sets
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DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
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