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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
Homological construction
introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects
Derived_category
Construction in category theory
so does CI, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted R n lim ← : C I
Inverse_limit
Tool to track locally defined data attached to the open sets of a topological space
only defined on the level of derived categories, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If
Sheaf_(mathematics)
Mapping between categories
In category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as
Functor
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
Branch of mathematics
for fixed A in ModR. This is a left exact functor and thus has right derived functors RnT. The Ext functor is defined by Ext R n ( A , B ) = ( R n T
Homological_algebra
Construction in homological algebra
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological
Ext_functor
Functor between abelian categories
morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related
Delta-functor
In mathematics, a mapping between categories
In mathematics, the direct image functor describes how structured data assigned to one space can be systematically transferred to another space using
Direct_image_functor
Category theory constructs
as) a Kan extension from 1956 was in homological algebra to compute derived functors. In Categories for the Working Mathematician, Saunders Mac Lane titled
Kan_extension
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
Formalism in homological algebra
operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. the direct image
Six_operations
Special kind of adjunction between categories named after Daniel Quillen
total derived functor theorem of Quillen says that the total left derived functor LF: Ho(C) → Ho(D) is a left adjoint to the total right derived functor RG:
Quillen_adjunction
Mathematical concept
Formally, a diagram of shape J {\displaystyle J} in C {\displaystyle C} is a functor from J {\displaystyle J} to C {\displaystyle C} : F : J → C . {\displaystyle
Limit_(category_theory)
Generalization of (co)homology using chain complexes
derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor R
Hyperhomology
Spectral sequence
computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\displaystyle
Grothendieck spectral sequence
Grothendieck_spectral_sequence
Tools for studying groups based on techniques from algebraic topology
exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by H n ( G ,
Group_cohomology
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
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
Relate the direct image and the pull-back of sheaves
{F}})} in the derived category of sheaves on S', similarly as mentioned above. Here L g ∗ {\displaystyle Lg^{*}} is the (total) derived functor of the pullback
Base_change_theorems
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
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
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
adjoint functors, as does the unicity. The notation Rf! is an abuse of notation insofar as there is in general no functor f! whose derived functor would
Exceptional inverse image functor
Exceptional_inverse_image_functor
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
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
Algebraic structure associated with a topological space
formulate homology theories as derived functors on appropriate abelian categories, measuring the failure of an appropriate functor to be exact. One can describe
Homology_(mathematics)
a resolution of a chain complex. It can be used to construct hyper-derived functors. It is named in honor of Henri Cartan and Samuel Eilenberg. Let A {\displaystyle
Cartan–Eilenberg_resolution
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
the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor − ⊗ A − : M A × A M → R M {\displaystyle
Derived_tensor_product
Category in mathematics
additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as
Triangulated_category
Construction in homological algebra
mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central
Tor_functor
Branch of mathematics
intersection theory has been developed. The term "derived" is used in the same way as derived functor or derived category, in the sense that the category of
Derived_algebraic_geometry
In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a
Čech-to-derived functor spectral sequence
Čech-to-derived_functor_spectral_sequence
Construct in algebraic geometry
of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this definition
Cotangent_complex
Exact sequence used to describe the structure of an object
the derived functors RiF(En) vanish for all i > 0 and n ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors
Resolution_(algebra)
Tool in algebraic topology
the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for
Sheaf_cohomology
Algebraic structure used in topology
functors from the derived category of sheaves on X to abelian groups. In a broad sense of the word, "cohomology" is often used for the right derived functors
Cohomology
Characterizing property of mathematical constructions
in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal
Universal_property
Sheaf cohomology on the étale site
derived functors of left exact functors. The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors
Étale_cohomology
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
Mathematical object in sheaf cohomology
construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied
Injective_sheaf
Mathematical category
defined and what is derived. A logical functor is a functor between topoi that preserves finite limits and power objects. Logical functors preserve the structures
Topos
n f ∗ {\displaystyle R^{n}f_{*}} is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of f − 1 ( U )
Decomposition theorem of Beilinson, Bernstein and Deligne
Decomposition_theorem_of_Beilinson,_Bernstein_and_Deligne
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
Sheaf consisting of modules on a ringed space; generalizing vector bundles
{\displaystyle \operatorname {H} ^{i}(X,-)} as the i-th right derived functor of the global section functor Γ ( X , − ) {\displaystyle \Gamma (X,-)} . Given a ringed
Sheaf_of_modules
Tool in homological algebra
algebraic situations involving derived functors. While their theoretical importance has decreased since the introduction of derived categories, they are still
Spectral_sequence
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 sequence
\mathrm {Sh_{Ab}} (Y)\xrightarrow {\Gamma } \mathrm {Ab} .} Thus the derived functors of Γ ∘ f ∗ {\displaystyle \Gamma \circ f_{*}} compute the sheaf cohomology
Leray_spectral_sequence
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
Smallest normal subgroup by which the quotient is commutative
G])\subseteq [H,H]} . This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below
Commutator_subgroup
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
an incorrect theorem about the vanishing of the first derived functor of the inverse limit functor under certain general conditions. However, in 2002, Amnon
List_of_incomplete_proofs
Duality for sheaves of k-modules over a locally compact space
finiteness conditions discussed below) certain derived image functors for sheaves are actually adjoint functors. There are two versions. Global Verdier duality
Verdier_duality
Generalisations of Serre duality in mathematics
proper (compact) support; they are bundled up into a single functor by means of the derived category formulation of homological algebra (introduced with
Coherent_duality
French mathematician (1928–2014)
theory for schemes X over a base field k Delta-functor – Functor between abelian categories Derivator Derived category – Homological construction Descent
Alexander_Grothendieck
Quillen functor M → L C M {\displaystyle M\to L_{C}M} whose left derived functor sends all morphisms in C to weak equivalences. Any left Quillen functor M →
Bousfield_localization
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
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
of the spectrum form a closed subset. Ext The Ext functors, the derived functors of the Hom functor. extension 1. An extension of an ideal is the ideal
Glossary of commutative algebra
Glossary_of_commutative_algebra
Topic in abstract algebra
It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis
Tilting_theory
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
geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each
Functor represented by a scheme
Functor_represented_by_a_scheme
paracompact), the derived-functor cohomology agrees with this Čech cohomology obtained by direct limits. However, like the derived functor cohomology, this
Leray_cover
Design pattern in functional programming to build generic types
as a functor: map : (a → b) → (ma → mb) This is not always a major issue, however, especially when a monad is derived from a pre-existing functor, whereupon
Monad (functional programming)
Monad_(functional_programming)
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
Theory of cohomology for commutative rings
B-module. The André–Quillen cohomology groups are the derived functors of the derivation functor DerA(B, M). Before the general definitions of André and
André–Quillen_cohomology
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 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
Construction 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 in category
Cone_(category_theory)
Topics referred to by the same term
Torsion group, in group theory and arithmetic geometry Tor functor, the derived functors of the tensor product of modules over a ring Torsion-free module
Torsion
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
Existence and uniqueness theorem for certain partial differential equations
the non homogeneous parts of each equation and the vanishing of a derived functor E x t 1 {\displaystyle Ext^{1}} . Let n ≤ m {\displaystyle n\leq m}
Cauchy–Kovalevskaya_theorem
Relates the homology of two objects to the homology of their product
This correction factor is expressed in terms of the Tor functor, the first derived functor of the tensor product. When R is a PID, then the correct statement
Künneth_theorem
Category-theoretic construction
Thus the contravariant hom-functor changes coproducts into products. Stated another way, the hom-functor, viewed as a functor from the opposite category
Coproduct
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
On representability of a contravariant functor on the category of connected CW complexes
contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor. More specifically
Brown's representability theorem
Brown's_representability_theorem
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
Cohomology theory for Lie algebras
(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor M ↦ M g :=
Lie_algebra_cohomology
Transformations induced by a mathematical group
coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants. Given g in G and x in X with g⋅x = x, it is
Group_action
History of maths
Verdier Triangulated categories and triangulated functors. Derived categories and derived functors are special cases of these 1963 Jim Stasheff A∞-algebras:
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
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
Theorem in category theory
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Lawvere's_fixed-point_theorem
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
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
Concept in category theory
homology of E ( F U ∗ M ) {\displaystyle E(FU_{*}M)} is the n-th left derived functor of E evaluated at M; i.e., Tor n R ( M , N ) {\displaystyle \operatorname
Cotriple_homology
Topic in mathematics
sequence of the composition of two derived functors. Indeed, H ∗ ( G , − ) {\displaystyle H^{*}(G,-)} is the derived functor of ( − ) G {\displaystyle (-)^{G}}
Lyndon–Hochschild–Serre spectral sequence
Lyndon–Hochschild–Serre_spectral_sequence
Type of vector space
Langlands correspondence. The derived Hecke algebra is a further generalization of Hecke algebras to derived functors. It was introduced by Peter Schneider
Hecke_algebra
Generalization of algebraic variety
same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules. Flat
Scheme_(mathematics)
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
This kind of functor is often called a dualizing functor. A celebrated theorem of Mukai states that there is an isomorphism of derived categories D b
Dual_abelian_variety
Indexed collection of objects and morphisms in a category
equivalently, a functor from a fixed index category to some category. Formally, a diagram of type J in a category C is a (covariant) functor D : J → C. The
Diagram_(category_theory)
Module over a sheaf of differential operators
while its pullback is a left module over X. This functor is right exact, its left derived functor is denoted Lf∗. Conversely, for a right DX-module N
D-module
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
Theorem in algebra
conceptually explained using the language of adjoint functors and derived categories: the functor between the derived categories of R- and k-modules induced by regarding
Matlis_duality
Abstract homotopical model for topological spaces
consider not only an abelian category, but also its derived category. A higher local system is then an ∞-functor L ∙ : Π ∞ X → D ( Ab ) {\displaystyle {\mathcal
∞-groupoid
Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level
Cohomology_operation
this way, cyclic homology (and cohomology) may be interpreted as a derived functor, which can be explicitly computed by the means of the (b, B)-bicomplex
Cyclic_homology
DERIVED FUNCTOR
DERIVED FUNCTOR
Girl/Female
Indian
Derived from kamadhenu
Girl/Female
Muslim
Derived from zarwari
Girl/Female
Tamil
Derived from kamadhenu
Boy/Male
British, English
Derived from Cerdic
Female
Hebrew
(×”ï‹×©××¢-× ×) Hebrew unisex name derived from hosha'na, HOSHA'NA means "deliver us."Â
Girl/Female
Australian, French
Derived from Lorraine
Girl/Female
Tamil
Victoria | விகà¯à®Ÿà¯‹à®°à®¿à®¯à®¾Â Â
Derived from Victoria triumphant
Victoria | விகà¯à®Ÿà¯‹à®°à®¿à®¯à®¾Â Â
Girl/Female
Australian, French
Derived from Lorraine
Girl/Female
Bengali, Indian, Kannada, Marathi
Derived from Ariana
Boy/Male
Tamil
Derived from Lord Shiva
Boy/Male
Arabic, Muslim
Derived from Abraham
Girl/Female
Indian
Derived from gulwari
Girl/Female
Indian
Derived from zarwari
Girl/Female
Greek
The sea nymphs.
Girl/Female
Spanish
derived from John.
Girl/Female
Muslim
Derived from gulwari
Girl/Female
Spanish
derived from John.
Girl/Female
Arabic, Muslim
Derived from Sarah
Surname or Lastname
English
English : occupational name for a driver of horses or oxen attached to a cart or plow, or of loose cattle, from a Middle English agent derivative of Old English drīfan ‘to drive’.
Girl/Female
Christian, German
Derived from Lorraine
DERIVED FUNCTOR
DERIVED FUNCTOR
Girl/Female
Hindu
Proficient, Magical, An aspirant, Seeker
Girl/Female
Hindu, Indian, Marathi
Daughter of Fire
Boy/Male
Tamil
Lord Krishna
Girl/Female
Arabic
Melody; Plural of Nagham
Boy/Male
Indian, Punjabi, Sikh
Lamp of the World
Boy/Male
Muslim
Bounty of my Lord
Girl/Female
Italian
Youthful.
Girl/Female
Indian
Noble, Honorable
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from Evreux in Eure, France, probably named from its association with the Eburovices, a Gaulish tribe.
Boy/Male
Hindu
An ancient ruler
DERIVED FUNCTOR
DERIVED FUNCTOR
DERIVED FUNCTOR
DERIVED FUNCTOR
DERIVED FUNCTOR
a.
Free from ambiguity; unequivocal; unmistakable; unquestionable; clear; evident; as, a decided advantage.
n.
One who derives.
v. t.
To trace the origin, descent, or derivation of; to recognize transmission of; as, he derives this word from the Anglo-Saxon.
a.
Driven to the end, as a nail; driven close.
n.
One who, or that which, deprives.
v. t.
To give forth in action or exercise; to discharge; as, to deliver a blow; to deliver a broadside, or a ball.
p. p.
of Drive. Also adj.
imp. & p. p.
of Deprive
imp. & p. p.
of Deride
a.
Having nerves of a special character; as, weak-nerved.
v. t.
To obtain one substance from another by actual or theoretical substitution; as, to derive an organic acid from its corresponding hydrocarbon.
v. t.
To impel or urge onward by force in a direction away from one, or along before one; to push forward; to compel to move on; to communicate motion to; as, to drive cattle; to drive a nail; smoke drives persons from a room.
a.
Corrupted or depraved by one's self.
p. p.
Driven.
imp. & p. p.
of Derive
a.
Driven from a house; deprived of shelter.
a.
Devised by one's self.
n.
One who derides, or laughs at, another in contempt; a mocker; a scoffer.
a.
Free from doubt or wavering; determined; of fixed purpose; fully settled; positive; resolute; as, a decided opinion or purpose.
p. p.
of Drive