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Mathematical object in category theory
homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives). In the category
Injective_object
Mathematical object in abstract algebra
in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure
Injective_module
Notion in abstract algebra
mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest
Injective_hull
Type of object in category theory
homological algebra. The dual notion of a projective object is that of an injective object. An object P {\displaystyle P} in a category C {\displaystyle
Projective_object
Abelian group in which every element can, in some sense, be divided by positive integers
Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary
Divisible_group
Function that preserves distinctness
{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If
Injective_function
Mathematical object in sheaf cohomology
the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;
Injective_sheaf
Homological construction
enough injectives. Embedding X into some injective object I0, the cokernel of this map into some injective I1 etc., one constructs an injective resolution
Derived_category
Exact sequence used to describe the structure of an object
flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a module M {\displaystyle
Resolution_(algebra)
Type of metric space
short map) on a bounded injective space has a fixed point. A metric space is injective if and only if it is an injective object in the category of metric
Injective_metric_space
Homological construction in category theory
I^{0}\to I^{1}\to I^{2}\to \cdots } where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence
Derived_functor
Spectral sequence
lemma: Lemma—If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n, H n
Grothendieck spectral sequence
Grothendieck_spectral_sequence
Pure-injective modules in mathematics
algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact,
Algebraically_compact_module
Inclusion of one mathematical structure in another, preserving properties of interest
When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-preserving
Embedding
submodule of the codomain. An injective hull of an object A is an essential monomorphism from A to an injective object. Hashimoto, Mitsuyasu (November
Essential_monomorphism
Programming technique of receiving dependencies
Instead, the receiving "client" (object or function) is provided with its dependencies by external code (an "injector"), which it is not aware of. Dependency
Dependency_injection
Category whose objects are sets and whose morphisms are functions
monomorphisms are the injective maps, and the isomorphisms are the bijective maps. The empty set serves as the initial object in Set with empty functions
Category_of_sets
Mathematical object
Hopfian objects and cohopfian objects have an elementary interaction with projective objects and injective objects. The two results are: An injective hopfian
Hopfian_object
Type of Abelian category (in category theory in mathematics)
categories. Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the
Grothendieck_category
Software development methodology
means if an object depends upon having an instance of some other object, then the needed object is "injected" into the dependent object (for example
Object-oriented analysis and design
Object-oriented_analysis_and_design
Philosophy terms referring to an observer versus the thing observed
subject is any individual, person, or observer. Also in philosophy, an object is any of the things observed or experienced by a subject, which may even
Subject and object (philosophy)
Subject_and_object_(philosophy)
Category whose objects are rings and whose morphisms are ring homomorphisms
Ring are the injective epimorphisms. The inclusion Z → Q is an example of a bimorphism which is not an isomorphism. The only injective object in Ring up
Category_of_rings
category with enough injectives is the least non-negative integer n such that every object in the category admits an injective resolution of length at
Glossary_of_category_theory
Structure-preserving map between two algebraic structures of the same type
then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x
Homomorphism
Object within another object of the same category
quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous
Subobject
projective cover of an object M is in a sense the best approximation of M by a projective object P. Projective covers are the dual of injective envelopes. Let
Projective_cover
Functors which are surjective and injective on hom-sets
faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range
Full_and_faithful_functors
If there are more items than boxes holding them, one box must contain at least two items
A to B that is not injective, then no surjection from A to B is injective. In fact no function of any kind from A to B is injective. This is not true for
Pigeonhole_principle
Abelian categories, while abstractly defined, are in fact concrete categories of modules
of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules. Let L ⊂ Fun ( A , A b
Mitchell's_embedding_theorem
Lemma in category theory about commutative diagrams
p are injective and l is surjective. Let c in C be such that n(c) = 0. t(n(c)) is then 0. By commutativity, p(h(c)) = 0. Since p is injective, h(c) =
Five_lemma
Injective homomorphism
morphisms h : Z → X, is injective for all objects Z. Every morphism in a concrete category whose underlying function is injective is a monomorphism; in
Monomorphism
Category whose objects are metric spaces and whose morphisms are metric maps
there are no zero objects in Met. The injective objects in Met are called injective metric spaces. Injective metric spaces were introduced and studied
Category_of_metric_spaces
Generalization of (co)homology using chain complexes
arrows, replacing injective objects with projective ones, and so on. Suppose that A is an abelian category with enough injectives and F a left exact
Hyperhomology
Distant planetesimals in the Solar System
Kuiper belt, the scattered disc and the detached objects—three nearer reservoirs of trans-Neptunian objects. Both regions lie well beyond the heliosphere
Oort_cloud
an Abelian category for example. Projective generators (and their dual injective cogenerators) are often very powerful tools to have when doing algebra
Generator_(category_theory)
3D model's surface projected to a 2D image
respective face of polygon. UV mapping may use repeating textures, or an injective 'unique' mapping as a prerequisite for baking. For any point P {\displaystyle
UV_mapping
Process of deriving classes from, and organizing them into, a hierarchy
In object-oriented programming, inheritance is the mechanism of basing an object or class upon another object (prototype-based inheritance) or class (class-based
Inheritance (object-oriented programming)
Inheritance_(object-oriented_programming)
Computer bug exploit caused by invalid data
validation, and other approaches to help mitigate the risk of an attack. Using object-relational mapping can further help prevent users from directly manipulating
Code_injection
Mathematical sequence
enough injectives, F {\displaystyle F} a left-exact functor, and G {\displaystyle G} sending injective objects to F {\displaystyle F} -acyclic objects, then
Leray_spectral_sequence
Theorem on extension of bounded linear functionals
category-theoretic terms, the underlying field of the vector space is an injective object in the category of locally convex vector spaces. On a normed (or seminormed)
Hahn–Banach_theorem
Injective polynomial functions are bijective
theorem is often given as this special case: If P {\displaystyle P} is an injective polynomial function from an n {\displaystyle n} -dimensional complex vector
Ax–Grothendieck_theorem
Operation on mathematical functions
0. The picture shows another example. The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective)
Function_composition
Association of one output to each input
function f is injective (or one-to-one, or is an injection) if f(a) ≠ f(b) for every two different elements a and b of X. Equivalently, f is injective if and
Function_(mathematics)
Algebraic structure with "nice" duality properties
and right self-injective. For a field k, a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right A-module Homk(A
Frobenius_algebra
Internet error message
page view. Many organizations use 404 error pages as an opportunity to inject humor into what may otherwise be a serious website. For example, Metro UK
HTTP_404
Map (arrow) between two objects of a category
inverse is injective, and a morphism that is injective is a monomorphism. In concrete categories, monomorphisms are often, but not always, injective; thus
Morphism
Mathematical function such that every output has at least one input
second kind. A non-injective surjective function (surjection, not a bijection) An injective surjective function (bijection) An injective non-surjective function
Surjective_function
Branch of mathematics that studies algebraic structures
module Projective cover Swan's theorem Quillen–Suslin theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated
List of abstract algebra topics
List_of_abstract_algebra_topics
having enough injectives and an additive (covariant) functor F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} , an acyclic object with respect
Acyclic_object
Finding the number of elements of a finite set
same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as
Counting
Systematic classification of 12 related enumerative problems concerning two finite sets
X is equivalent to counting injective functions N → X. Counting n-combinations of X is equivalent to counting injective functions N → X up to permutations
Twelvefold_way
Overview of and topical guide to category theory
of magmas Initial object Terminal object Zero object Subobject Group object Magma object Natural number object Exponential object Epimorphism Monomorphism
Outline_of_category_theory
One-to-one correspondence
complex plane. An injective non-surjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function
Bijection
3-term complex A → B → C of objects in some abelian category whose middle term B is projective, whose first map A → B is injective, and whose second map B → C
Monad_(homological_algebra)
properties concern the domain, the codomain and the image of functions. Injective function: has a distinct value for each distinct input. Also called an
List_of_types_of_functions
Functor that preserves short exact sequences
is a contravariant left-exact functor; it is exact if and only if A is injective. If k is a field and V is a vector space over k, we write V * = Homk(V
Exact_functor
Structure-preserving function between two rings
itself. Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then
Ring_homomorphism
Sheaf cohomology on the étale site
The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors.
Étale_cohomology
Self-self morphism
Epimorphism (surjective homomorphism) Frobenius endomorphism Monomorphism (injective homomorphism) Lang. Algebra. p. 10. Lang. Algebra. p. 54. Jacobson (2009)
Endomorphism
Concept in mathematics
injective hull of M. The injective hull is necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the
Essential_extension
{\displaystyle \operatorname {Fun} ({\mathcal {I}},{\mathcal {C}})} . Injective cofibrations and injective weak equivalences are the natural transformations, which
Injective and projective model structure
Injective_and_projective_model_structure
Video game series
creatures by a learning computer (for verbs) or by repeating the name of the object while the creature looks at it. Once a creature understands language, the
Creatures_(video_game_series)
Elements taken to zero by a homomorphism
whether a homomorphism is injective. In these cases, the kernel is a congruence relation. Kernels allow defining quotient objects (also called quotient algebras
Kernel_(algebra)
Topics referred to by the same term
Project (disambiguation) Proform, which covers proadjective Adjective Injective Surjective This disambiguation page lists articles associated with the
Projective
Reusable solution template to a commonly-needed software behavior
the problem they are trying to solve, and object-oriented patterns are not necessarily suitable for non-object-oriented languages. Patterns originated as
Software_design_pattern
Smallest complete lattice containing a partial order
complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of S is the injective hull of S. Several researchers
Dedekind–MacNeille_completion
Endomorphism algebra of an abelian group
not contain any non-trivial idempotent elements. If the module is an injective module, then indecomposability is equivalent to the endomorphism ring
Endomorphism_ring
Theorem in algebra
This right adjoint sends the injective hull E ( k ) {\displaystyle E(k)} mentioned above to k, which is a dualizing object in D ( k ) {\displaystyle D(k)}
Matlis_duality
Medical facility
They are usually in areas where there is an open drug scene and where injecting in public places is common. The primary target group for DCR services
Supervised_injection_site
Computer hacking technique
database schema or inject malicious scripts. For example, the following permissions restrict a database user from accessing system objects:[citation needed]
SQL_injection
Object-oriented programming language
Smalltalk is a purely object-oriented programming language that was originally created in the 1970s for educational use, specifically for constructionist
Smalltalk
program. The table below does not include lesser Apollo mission artificial objects, such as a hammer and other tools, retroreflectors, Apollo Lunar Surface
List of artificial objects on the Moon
List_of_artificial_objects_on_the_Moon
Sexual arousal a person receives from an object or situation
fixation on anything not considered sexual by its respective nature. The object of interest is called the fetish; the person who has a fetish is a fetishist
Sexual_fetishism
Human disease
parapoxvirus. It can occur in humans who handle infected animals or contaminated objects. One-third of cases may develop erythema multiforme. Once resolved, a person
Orf_(disease)
Category whose objects are abelian groups and whose morphisms are group homomorphisms
prototypical example of a Grothendieck category. An object in A b {\displaystyle \mathbf {Ab} } is injective if and only if it is a divisible group; it is projective
Category_of_abelian_groups
Shared boundary between elements of a computing system
type definition; anywhere an object can be exchanged (for example, in a function or method call) the type of the object to be exchanged can be defined
Interface_(computing)
1961 Soviet medium tank
modernization of the T-62 has now been spotted twice in the war; This is Object 169, a modernization which preceded the main T-62M. Only very few vehicles
T-62
Differentiable function whose derivative is everywhere injective
everywhere injective. Explicitly, f : M → N is an immersion if D p f : T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function
Immersion_(mathematics)
Mathematical subcategories of Grothendieck categories
category. An object X {\displaystyle X} in B {\displaystyle {\mathcal {B}}} is injective if and only if i ( X ) {\displaystyle i(X)} is injective in A {\displaystyle
Giraud_subcategory
Category whose objects and morphisms are inside a bigger category
faithful functor that is injective on objects. Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F {\displaystyle
Subcategory
Function that returns its argument unchanged
{\displaystyle X} . The identity function on X {\displaystyle X} is clearly an injective function as well as a surjective function (its codomain is also its range)
Identity_function
History of maths
and an inflation x→I(x) (the injective hull of x) such that both P(x) and I(x) are in the category of pro/injective objects. A Frobenius category E is an
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Algebraic structure with addition and multiplication
{\overline {f}}} is the polynomial function defined by f. The resulting map is injective if and only if R is infinite. Given a non-constant monic polynomial f
Ring_(mathematics)
Sheaf theory concept
categorical arguments coming from the properties of injective objects. It is also possible to define an injective resolution which is canonical in the sense that
Godement_resolution
Depending on this, the distribution can be: Injective distribution: every box must have at most 1 object ( k ≤ n {\displaystyle k\leq n} ) Surjective
Combinatorial_modelling
Mathematical set of all subsets of a set
while 0 means it does not. For the whole power set of S, we get: Such an injective mapping from P(S) to integers is arbitrary, so this representation of
Power_set
Class used for injecting methods
In object-oriented programming languages, a mixin (or mix-in) is a class that contains methods for use by other classes without having to be the parent
Mixin
provides, on top of this visitor API, a tree API that represents classes as object constructs. Both APIs can be used for modifying the binary bytecode, as
ObjectWeb_ASM
Describing something mathematical with variables
values can refer to the same point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ, φ, z) and
Parametrization_(geometry)
"size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and
Accessible_category
Method of introducing a drug
talc in the drugs of people who inject substances. More commonly, the inflammatory response to these foreign objects causes granulation tissue to form
Drug_injection
Mathematical metric
induced by the supremum norm or uniform norm. It is an example of an injective metric. In two dimensions, i.e. plane geometry, if the points a and b
Chebyshev_distance
French mathematician (1933–2015)
quasi-coherent sheaves on X, by considering the spectrum of indecomposable injective objects in the category. This theorem, later vastly generalized by Alexander
Pierre_Gabriel
Representation of groups by permutations
{Sym} (G)} sending each element g to ℓ g {\displaystyle \ell _{g}} is an injective homomorphism, so it defines an isomorphism from G onto a subgroup of Sym
Cayley's_theorem
Class of web security vulnerabilities
to overwrite arbitrary properties in an object's prototype. In a prototype pollution attack, attackers inject properties into existing JavaScript construct
Prototype_pollution
least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps
Simplicial_presheaf
Criminal practice of stealing electrical energy
station sent a power surge into the line to burn out or destroy foreign objects trespassing on the line. There are various types of electrical power theft
Electricity_theft
vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 Meaning of “efface” in “effaceable functor” and “injective effacement” v t e
Effaceable_functor
Ethnic cleansing of Palestinians
pro-Zionist stand. In addition, according to the Arabs, the committee's final object – the partition – was pre-decided by the Americans. According to this opinion
Nakba
encounters, claimed or speculated to be associated with unidentified flying objects (UFOs) (not to be confused with the meaning of the term "alien species"
List of alleged extraterrestrial beings
List_of_alleged_extraterrestrial_beings
INJECTIVE OBJECT
INJECTIVE OBJECT
Surname or Lastname
English
English : nickname from the adjective bony, denoting a scrawny individual with prominent bones.
Surname or Lastname
French
French : metonymic occupational name for a gardener, from the objective case (gard) of Old French gardin ‘garden’.English : variant spelling of Guard.Norwegian : habitational name from a farmstead so named, from Old Norse garðr ‘farm’.Swedish (Gård) : topographic or ornamental name from gård ‘farm’.
Girl/Female
Hindu, Indian
Queen of Horizon; Injection
Surname or Lastname
English
English : nickname for a person with a ruddy complexion, from an adjective derivative of Middle English mad(d)er ‘madder’, the dye plant (see Mader 1), here used in a transferred sense.
Boy/Male
Muslim
Objective, Goal
Surname or Lastname
English
English : habitational name from places in North and West Yorkshire named Barden, from Old English bere ‘barley’ (or the derived adjective beren) + denu ‘valley’.
Boy/Male
Arabic, Muslim
Wanted; Unknown; Objective; Goal
Girl/Female
African, American, Australian, British, Chinese, Danish, English, Hebrew, Indian, Irish, Japanese, Swahili
Intention; Female Champion; Aim; Objective; Goal; Purpose; Beauty; Brightness; God Gifted
Boy/Male
Hindu, Indian, Sanskrit
Adjective Devil
Girl/Female
Indian
An Adjective to Happy as Happiest; A Type of Grass that Cuts Only Bad Person's Skin
Girl/Female
Latin American Spanish
Dazzling white. Bright, glowing white. Derived from the feminine of the Latin adjective meaning...
Boy/Male
Hindu
Valin means courage in sanskrit. adding i hence Valini would keep the meaning the same as but make it feminine as in Hindi An i at the end of a noun or adjective makes it feminine
Surname or Lastname
English
English : from the informal England adjective Brummagem ‘of or relating to Birmingham’, hence a habitational name for someone from the city of Birmingham in the West Midlands.
Surname or Lastname
English
English : occupational name for an officer of a court of justice, from the English vocabulary word bailiff, which is from the objective case of Old French bailis (see Bayliss).
Boy/Male
Muslim/Islamic
Objective goal
Surname or Lastname
Irish
Irish : reduced Anglicized form of Gaelic Ó Teimhin ‘descendant of Teimhean’, from teimhean ‘dark’, an adjective from teimhe ‘dusk’, ‘darkness’.English : probably a habitational name for someone from Tyneside in northeast England.
Surname or Lastname
Americanized form of German, Dutch, or northern French Happe.English
Americanized form of German, Dutch, or northern French Happe.English : nickname from the adjective happy.
Female
Greek
(ΑἰκατεÏίνη) Greek name of uncertain etymology, but from an early date it has been associated with the Greek adjective katharos, AIKATERINE means "pure."Â
Boy/Male
Indian
Objective, Goal
Boy/Male
Tamil
Valin means courage in sanskrit. adding i hence Valini would keep the meaning the same as but make it feminine as in Hindi An i at the end of a noun or adjective makes it feminine
INJECTIVE OBJECT
INJECTIVE OBJECT
Girl/Female
Hindu
Decorative, Name of a river
Boy/Male
Tamil
Palashranjan | பலாஷரஂஜந
Beautiful like a Palash
Girl/Female
African, Danish, German, Kenyan, Netherlands
Beautiful
Boy/Male
Afghan, African, Arabic, Bengali, French, German, Hindu, Indian, Malayalam, Marathi, Muslim, Pashtun, Sindhi, Turkish
Holy; Chaste; Modest; Clean; Pure; Innocent; Unsullied; Untouched; From Muslim; Virtuous
Girl/Female
Tamil
Arshika | à®…à®°à¯à®·à¯€à®•à®¾
Who gives happiness
Boy/Male
Tamil
Brilliance
Girl/Female
Australian, Dutch, Finnish, German, Hebrew, Swedish
Gift from God; God is Merciful
Boy/Male
Australian, Irish
Dark
Boy/Male
Hindu, Indian
Amazingly
Girl/Female
Indian, Malayalam
Green
INJECTIVE OBJECT
INJECTIVE OBJECT
INJECTIVE OBJECT
INJECTIVE OBJECT
INJECTIVE OBJECT
a.
Not active; inert; esp., not exhibiting any action or activity on polarized light; optically neutral; -- said of isomeric forms of certain substances, in distinction from other forms which are optically active; as, racemic acid is an inactive tartaric acid.
n.
An object glass. See under Object, n.
a.
Pertaining to, or designating, the case which follows a transitive verb or a preposition, being that case in which the direct object of the verb is placed. See Accusative, n.
a.
Leading to inferences; proceeding by, derived from, or using, induction; as, inductive reasoning.
a.
Not active; having no power to move; that does not or can not produce results; inert; as, matter is, of itself, inactive.
n.
A specimen prepared by injection.
n.
A word used with a noun, or substantive, to express a quality of the thing named, or something attributed to it, or to limit or define it, or to specify or describe a thing, as distinct from something else. Thus, in phrase, "a wise ruler," wise is the adjective, expressing a property of ruler.
n.
The act of injecting or throwing in; -- applied particularly to the forcible throwing in of a liquid, or aeriform body, by means of a syringe, pump, etc.
a.
Beginning; expressing or indicating beginning; as, an inceptive proposition; an inceptive verb, which expresses the beginning of action; -- called also inchoative.
n.
Same as Objective point, under Objective, a.
v. t.
To make an adjective of; to form or change into an adjective.
a.
Able and apt to invent; quick at contrivance; ready at expedients; as, an inventive head or genius.
n.
An expression which inveighs or rails against a person; a severe or violent censure or reproach; something uttered or written, intended to cast opprobrium, censure, or reproach on another; a harsh or reproachful accusation; -- followed by against, having reference to the person or thing affected; as an invective against tyranny.
n.
An inceptive word, phrase, or clause.
a.
Facilitating induction; susceptible of being acted upon by induction; as certain substances have a great inductive capacity.
a.
Operating by induction; as, an inductive electrical machine.
n.
The objective case.
a.
Capable of, or pertaining to, inflection; deflecting; as, the inflective quality of the air.
a.
Not disposed to action or effort; not diligent or industrious; not busy; idle; as, an inactive officer.
n.
Added to a substantive as an attribute; of the nature of an adjunct; as, an adjective word or sentence.