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Mapping between categories
functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group)
Functor
Mathematical group formed from the automorphisms of an object
F:C_{1}\to C_{2}} is a functor mapping X 1 {\displaystyle X_{1}} to X 2 {\displaystyle X_{2}} , then F {\displaystyle F} induces a group homomorphism Aut
Automorphism_group
mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme
Group_functor
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
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 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
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
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
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
result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature
Yoneda_lemma
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
Functor mapping hom objects to an underlying category
between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category
Hom_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
Mathematical group of the homotopy classes of loops in a topological space
and therefore yields an isomorphism of their fundamental groups. The fundamental group functor takes products to products and coproducts to coproducts
Fundamental_group
Concept in mathematics
The natural functor from Lie groups or algebraic groups to Lie algebras can be factorized into a functor from Lie groups to formal group laws, followed
Formal_group_law
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
Operation in algebra and mathematics
a branch of mathematics, a monad is a triple (T,η,μ) consisting of a functor T from a category to itself and two natural transformations η,μ that satisfy
Monad_(category_theory)
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
Type of mathematical object
forgetful functor to sets is equivalent to the presheaf corresponding to G under the Yoneda embedding. (See also: group functor.) A homomorphism of group schemes
Group_scheme
Mathematical concept
forgetful functors. The free functor F : Set → Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and is,
Limit_(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
signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an abelian group. The signalizer functor theorem
Signalizer_functor
Concept in algebraic topology
singular homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups. A singular n-simplex in a topological
Singular_homology
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
Most general completion of a commutative square given two morphisms with same domain
Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest
Pushout_(category_theory)
Category whose objects are groups and whose morphisms are group homomorphisms
\mathbf {Grp} } the left adjoint functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U : G r p → S e t {\displaystyle
Category_of_groups
Certain functors from the category of modules over a fixed commutative ring to itself
especially in the field of representation theory, Schur functors (named after Issai Schur) are certain functors from the category of modules over a fixed commutative
Schur_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
Endofunctor on the category V of finite-dimensional vector spaces
In algebra, a polynomial functor is an endofunctor on the category 𝒱 of finite-dimensional vector spaces that depends polynomially on vector spaces.
Polynomial_functor
Duality for locally compact abelian groups
on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are
Pontryagin_duality
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
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)
Monoidal category
permutation representations of groups G which are profinite groups. The gist of the theory is that the fiber functor Φ of the Galois theory is replaced
Tannakian_formalism
Category whose objects are rings and whose morphisms are ring homomorphisms
respectively. Both of these functors have left adjoints. The left adjoint of A is the functor which assigns to every abelian group X (thought of as a Z-module)
Category_of_rings
consider (at least some special case of) an action of a group functor: viewing G as a functor, an action is given as a natural transformation satisfying
Group-scheme_action
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
Category whose objects are abelian groups and whose morphisms are group homomorphisms
forgetful functor A b → S e t {\displaystyle \mathbf {Ab} \to \mathbf {Set} } which assigns to each abelian group the underlying set, and to each group homomorphism
Category_of_abelian_groups
Transformations induced by a mathematical group
the zeroth cohomology group of G with coefficients in X, and the higher cohomology groups are the derived functors of the functor of G-invariants. Given
Group_action
Tools for studying groups based on techniques from algebraic topology
This functor is left exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they
Group_cohomology
Left adjoint to a forgetful functor to sets
that is equipped with a faithful functor to Set, the category of sets. Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that
Free_object
Category whose hom sets have algebraic structure
product as the monoidal operation (thinking of abelian groups as Z-modules). If there is a monoidal functor from a monoidal category M to a monoidal category
Enriched_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
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
Category with direct sums and certain types of kernels and cokernels
groups in the theory and its canonical nature. The concept of exact sequence arises naturally in this setting, and it turns out that exact functors,
Abelian_category
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
Construction in category theory
limit functor lim←:CI→C is left exact. If I is ordered (not simply partially ordered) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler
Inverse_limit
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
Mathematical category whose hom sets form Abelian groups
A and B of C, the function F:Hom(A,B)→Hom(F(A),F(B)) is a group homomorphism. Most functors studied between preadditive categories are additive. For a
Preadditive_category
Tool to track locally defined data attached to the open sets of a topological space
of abelian groups in the sense above. There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually
Sheaf_(mathematics)
Theorem classifying finite simple groups
theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.) A group is said to be of component
Classification of finite simple groups
Classification_of_finite_simple_groups
Topological concept in algebraic geometry
group is a functor: {Pointed algebraic varieties} → {Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or
Étale_fundamental_group
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
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
Type of category in mathematics
object, but now the set of morphisms is given by the group G {\displaystyle G} . Since any functor must give a G {\displaystyle G} -action on the target
Topos
Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering
Fiber_functor
Contravariant functor to Set
branch of mathematics, a presheaf on a category C {\displaystyle C} is a functor F : C o p → S e t {\displaystyle F\colon C^{\mathrm {op} }\to \mathbf {Set}
Presheaf_(category_theory)
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 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
Type of category in category theory
multiplying the degenerate matrices. A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in
Additive_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 considered as a discrete category. The definition
Product_(category_theory)
Construction in group theory
Zassenhaus groups. PGL(n, K) is an algebraic group of dimension n2 − 1 and an open subgroup of the projective space Pn2−1. As defined, the functor PSL(n,
Projective_linear_group
respect to some universe) and the morphisms functors. Fct(C, D), the functor category: the category of functors from a category C to a category D. Set, the
Glossary_of_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 {\displaystyle
Conservative_functor
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
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
Set of elements that commute with every element of a group
all of H unless f is surjective. Thus the center mapping G→Z(G) is not a functor between categories Grp and Ab, since it does not induce a map of arrows
Center_(group_theory)
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)
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
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 case of colimit in category theory
direct system over I is then the same as a covariant functor ℐ→𝒞. The colimit of this functor is the same as the direct limit of the original direct
Direct_limit
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
Group that is also a differentiable manifold with group operations that are smooth
transformation from the functor Lie to the identity functor on the category of Lie groups.) The exponential map from the Lie algebra to the Lie group is not always
Lie_group
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
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional
N-group_(category_theory)
In mathematics, invertible homomorphism
exist functors F:C→D and G:D→C which are mutually inverse to each other, that is, FG=1D (the identity functor on D) and GF=1C (the identity functor on C)
Isomorphism
Type of category in category theory
a special case of functor categories: every group can be considered as a one-object category, and G-sets are nothing but functors from this category
Cartesian_closed_category
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)
On the homotopy groups of the infinite symmetric product of a connected CW complex
its proof consists of showing that the composition of the homotopy group functors with the infinite symmetric product defines a reduced homology theory
Dold–Thom_theorem
Set of finitely supported functions from a group to a ring
algebra structure, the group algebra construction is left adjoint to the "group-like elements" functor; the following functors are an adjoint pair:
Group_ring
Abelian group extending a commutative monoid
monoid M to its Grothendieck group K. This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative
Grothendieck_group
Category
between preadditive categories that acts as a group homomorphism on each hom-set. Then it turns out that a functor between pre-abelian categories is left exact
Pre-abelian_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
How spheres of various dimensions can wrap around each other
differentials to the motivic sphere, and then use the Betti realization functor to push forward them to the classical sphere. Using this method, Isaksen
Homotopy_groups_of_spheres
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
coaugmented functor. Serre introduced the idea of working in homotopy theory modulo some class C of abelian groups. This meant that groups A and B were
Localization_of_a_category
Type of topological group in mathematics
functor induces an equivalence of categories LCAop → LCA. This functor exchanges several properties of topological groups. For example, finite groups
Locally_compact_group
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
construction of a scheme structure on (the representable functor version of) the Picard group, the Picard scheme, is an important step in algebraic geometry
Picard_group
Mathematics concept
the free group (similar to most constructions of free objects) is a functor from the category of sets to the category of groups. This functor is left adjoint
Free_group
Group homomorphism into the general linear group over a vector space
G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(x), the automorphism group of X. In the
Group_representation
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were
Zuckerman_functor
Area of mathematics
{\displaystyle e_{j}} . Alternating polynomials Symmetric polynomials Schur functor Robinson–Schensted correspondence Schur–Weyl duality Jucys–Murphy element
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Concept in mathematical category theory
category representing the group H in the same way. The requirement that F be a functor is then equivalent to specifying a group homomorphism φ:G→Aut(H)
Category_of_elements
Type of group in abstract algebra
representation theory of Lie groups, the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors. In the theory of
Symmetric_group
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
Mathematical functor in representation theory and algebraic topology
theory and algebraic topology, a Mackey functor is a type of functor that generalizes various constructions in group theory and equivariant homotopy theory
Mackey_functor
Smallest normal subgroup by which the quotient is commutative
The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups. The existence of
Commutator_subgroup
Category admitting tensor products
a strict monoidal category with the composition of functors as the product and the identity functor as the unit. Just like for any category E, the full
Monoidal_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
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
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 object
2-group
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