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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
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
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
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)
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
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
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
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
for any i. conservative functor A conservative functor is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful
Glossary_of_category_theory
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
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
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
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
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
Characterizing property of mathematical constructions
Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal
Universal_property
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
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)
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
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
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
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 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
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
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
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_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
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
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
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 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)
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 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-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
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
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)
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 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
Generalized object in category theory
the components and projections. If we regard this diagram as a functor, it is a functor from the index set I {\displaystyle I} considered as a discrete
Product_(category_theory)
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
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
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
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
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
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)
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
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)
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)
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
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
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
Quotient space of a codomain of a linear map by the map's image
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Cokernel
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
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
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
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)
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
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
Programming construct
In some languages, particularly C++, function objects are often called functors (not related to the functional programming concept). A typical use of a
Function_object
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 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 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)
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
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
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
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
Graphical representation of a morphism
and a monoidal functor to its underlying morphism of signatures, i.e. it forgets the identity, composition and tensor. The free functor C − : M o n S i
String_diagram
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
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
Applications of category theory
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Applied_category_theory
Injective homomorphism
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Monomorphism
Category in mathematical category theory
Heyting category, then e A {\displaystyle e_{\mathcal {A}}} is a conservative Heyting functor. A geometric category is a regular category which is well-powered
Coherent_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
Set of arguments where two or more functions have the same value
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Equaliser_(mathematics)
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)
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
an amnestic functor F : A → B is a functor for which an A-isomorphism ƒ is an identity whenever Fƒ is an identity. An example of a functor which is not
Amnestic_functor
Mathematical category with weak equivalences, fibrations and cofibrations
sets and simplicial commutative rings (given by the forgetful and free functors), and in nice cases one can lift model structures under an adjunction.
Model_category
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
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
Mathematical operation with two operands
Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound
Binary_operation
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
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
Study of categorified structures
consider quantum double groupoids to be fundamental groupoids defined via a 2-functor, which allows one to think about the physically interesting case of quantum
Higher-dimensional_algebra
Concept in homological algebra
triangulated category D {\displaystyle {\mathcal {D}}} with translation functor [ 1 ] {\displaystyle [1]} . A t-structure on D {\displaystyle {\mathcal
T-structure
Mathematical use of "for all"
quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential
Universal_quantification
Bi-universal property in category theory
in that category. Section 1.7 of Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics, vol. 39, Academic Press, ISBN 978-0-12-545150-5
Zero_morphism
Category theory concept
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Overcategory
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
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Tetracategory
Hypothesis in mathematical category theory
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Homotopy_hypothesis
stabilization of an ∞-category C having finite limits and base point is a functor from the stable ∞-category S to C. It preserves limits. The objects in
Stable_∞-category
In mathematics, collection of classes
Neumann–Bernays–Gödel set theory (NBG), and Morse–Kelley set theory (MK), admit non-conservative extensions that arise after adding a supplementary axiom of existence
Conglomerate_(mathematics)
\operatorname {Env} _{L}^{L}} can be defined as a functor. In the following list all envelopes can be defined as functors. 1. The completion X ▾ {\displaystyle
Envelope_(category_theory)
Application of homotopy to algebraic varieties
The collection { x ∗ } {\displaystyle \{x^{*}\}} is a conservative family of fibre functors for S h v ( S m S ) N i s {\displaystyle Shv(Sm_{S})_{Nis}}
A¹_homotopy_theory
application in algebraic geometry also known as base change) induce adjoint functors, which with the model structures can even become Quillen adjunctions. Let
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Aspect of category theory in mathematics
Concrete Forgetful functor Pre-abelian Preadditive Commutative diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence
Rig_category
homotopic. Weak inverses can always be assigned coherently: one can define a functor on any 2-group G that assigns a weak inverse to each object, so that each
2-group
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
Boy/Male
Arabic
Proof of Islam
Girl/Female
Hungarian, Indian, Sanskrit, Swahili
Snow; Excellent; Brave; Foreign; Stranger; Strange
Girl/Female
Assamese, Hindu, Indian, Sindhi, Traditional
Possessing Gold
Girl/Female
Irish
meaning pure.
Girl/Female
Hindu
Fair
Boy/Male
Tamil
Success in every work
Girl/Female
Arabic, Muslim
Petals of Flower
Girl/Female
Australian, Finnish, Latin
Mild; Soft; Silky
Boy/Male
Anglo, British, English
Lives on the Castle's Hill
Boy/Male
Tamil
Dhrupal | தà¯à®°à¯à®ªà®¾à®²
Prosperity with greenery, A area with full of greenery
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
CONSERVATIVE FUNCTOR
n.
The act of preserving, guarding, or protecting; the keeping (of a thing) in a safe or entire state; preservation.
a.
The quality of being conservative.
n.
A public place of instruction in any special branch, esp. music and the arts. [See Conservatory, 3].
a.
Observing; watchful.
a.
Relating to intercourse with men; social; -- opposed to contemplative.
n.
A member of the Conservative party.
n.
Conservation, as from injury, defilement, or irregular use.
a.
Of or pertaining to a political party which favors the conservation of existing institutions and forms of government, as the Conservative party in England; -- contradistinguished from Liberal and Radical.
a.
Tending or disposed to maintain existing institutions; opposed to change or innovation.
a.
Contentious; quarrelsome.
n.
One who, or that which, preserves from ruin, injury, innovation, or radical change; a preserver; a conserver.
a.
Having power to preserve in a safe of entire state, or from loss, waste, or injury; preservative.
n.
One who desires to maintain existing institutions and customs; also, one who holds moderate opinions in politics; -- opposed to revolutionary or radical.
n.
A dull old fellow; a person behind the times, over-conservative, or slow; -- usually preceded by old.
a.
Untractable; bigoted; obstinately and blindly or stupidly conservative.
n.
The principles of those adhering to the house of Bourbon; obstinate conservatism.
n.
The disposition and tendency to preserve what is established; opposition to change; the habit of mind; or conduct, of a conservative.
v. i.
To have mutual communication or intercourse by conservation.
a.
Having the power or quality of conservation.
n.
Excessive conservatism; hostility to progress.