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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
itself. The identity functor is an endofunctor. Diagonal functor The diagonal functor is defined as the functor from D to the functor category DC which sends
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
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)
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)
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
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
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
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
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
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
injection morphism to the l {\displaystyle l} -th component. Diagonal functor Diagonal embedding wikibooks:Category Theory/(Co-)cones and (co-)limits
Diagonal_morphism
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
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)
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
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)
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
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
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
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
Category whose hom sets have algebraic structure
usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory
Enriched_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
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
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
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
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
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
Mathematical category
the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf
Topos
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)
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
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
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
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
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
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
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
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
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)
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
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)
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
Mathematical object that generalizes the standard notions of sets and functions
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)
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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)
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
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
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
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
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
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
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)
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
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
Generalisation of a sheaf; a fibered category that admits effective descent
automorphisms which have been overcounted. A category c {\displaystyle c} with a functor to a category C {\displaystyle C} is called a fibered category over C {\displaystyle
Stack_(mathematics)
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
Abstract homotopical model for topological spaces
theorems about local systems is that they can be equivalently described as a functor from the fundamental groupoid Π X = Π ≤ 1 X {\displaystyle \Pi X=\Pi _{\leq
∞-groupoid
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
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
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
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
Injective homomorphism
Forgetful functor Elementary topos Grothendieck topos Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative
Monomorphism
Most general completion of a commutative square given two morphisms with same domain
we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when
Pushout_(category_theory)
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
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
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
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
Correspondence between properties of a category and its opposite
this context, the duality is often called Eckmann–Hilton duality. Adjoint functor Dual object Duality (mathematics) Opposite category Pulation square Jiří
Dual_(category_theory)
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 in C,
Conservative_functor
Generalization of algebraic spaces or schemes
associated functors). Moreover, this construction is functorial on ( S c h / S ) {\displaystyle (\mathrm {Sch} /S)} forming a contravariant 2-functor ( R (
Algebraic_stack
Applications of category theory
Forgetful functor Elementary topos Grothendieck topos Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative
Applied_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)
Surjective homomorphism
-)&\rightarrow &\operatorname {Hom} (X,-)\end{matrix}}} being a monomorphism in the functor category SetC. Every coequalizer is an epimorphism, a consequence of the
Epimorphism
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
{\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
Quotient space of a codomain of a linear map by the map's image
Forgetful functor Elementary topos Grothendieck topos Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative
Cokernel
Particular correspondence between two partially ordered sets
that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered
Galois_connection
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
Graph with oriented edges
characterizing the shape of, a representation V defined as a functor, specifically an object of the functor category FinVctKF(Q) where F(Q) is the free category
Directed_graph
Type of category in category theory
monoidal category. Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian closed categories. Cartesian
Cartesian_monoidal_category
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
Boy/Male
Arabic, French, Greek, Indian, Muslim
Star; The Flyer; Flying Eagle; Bird; Refers to a First Magnitude Star in the Constellation Lyra
Girl/Female
Hebrew Greek
or Elizabeth, from Elisheba, meaning either oath of God, or God is satisfaction. Also a...
Boy/Male
Indian
Chosen one, Another name of prophet Yaqub
Surname or Lastname
English and Scottish
English and Scottish : from the Old Norse personal name Ãsketill, composed of the elements áss ‘god’ + ketill ‘kettle’, ‘helmet’ (see Haskell). This name was in use both among Scandinavian settlers in northern England and among the Normans.
Boy/Male
Tamil
King of the empire
Boy/Male
Muslim
Slave of the one who is aware
Girl/Female
German, Hebrew, Indian, Latin, Sanskrit, Telugu
God; Sun; Bird; Strength; Desired; Cloud
Girl/Female
Tamil
Honored, Noble, Goddess Parvati
Girl/Female
Indian
Intelligent, Honest
Girl/Female
Indian, Modern
River
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
DIAGONAL FUNCTOR
a.
Diagonal; diametrical; hence; diametrically opposed.
adv.
In a diagonal direction.
a.
Pertaining to a decagon; having ten sides.
adv.
Diagonally.
a.
Pertaining to the shorter diagonal, as of a rhombic prism.
adv.
Diagonally laid, as tiles; ridgewise.
a.
Of or pertaining to a deacon.
a.
Cut slanting or diagonally, as cloth.
n.
A diagonal cloth; a kind of cloth having diagonal stripes, ridges, or welts made in the weaving.
n.
Diagonal braces sometimes fixed across the hold.
a.
Joining two not adjacent angles of a quadrilateral or multilateral figure; running across from corner to corner; crossing at an angle with one of the sides.
v. t.
To cut diagonally.
n.
The shorter of the diagonals in a rhombic prism.
n.
A member, in a framed structure, running obliquely across a panel.
a.
Notched in regular diagonal breaks; -- said of a line, or a bearing having such an edge.
a.
Having a single, distinct, diagonal cleavage; -- said of crystals.
n.
A right line drawn from one angle to another not adjacent, of a figure of four or more sides, and dividing it into two parts.
n.
A quadrilateral, one of whose diagonals is an axis of symmetry.
n.
A slant; a diagonal; as, to cut cloth on the bias.
a.
Diagonal.