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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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
Mathematical concept in category theory
quotient object. In the category of topological spaces, monomorphisms are precisely the injective continuous functions; but not all injective continuous
Subobject
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
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
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)
Theorem in combinatorics
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
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
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
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
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
Philosophy terms referring to an observer versus the thing observed
observer. Also in philosophy, an object is any of the things observed or experienced by a subject, physical objects such as chairs, clouds, and the Moon
Subject and object (philosophy)
Subject_and_object_(philosophy)
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
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
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)
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
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
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
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 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
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
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
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
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)
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
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
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
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
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
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
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
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
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)
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
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
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)
co-Hopfian if whenever φ : G → G {\displaystyle \varphi :G\to G} is an injective group homomorphism then φ {\displaystyle \varphi } is surjective, that
Co-Hopfian_group
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
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
Self-self morphism
Epimorphism (surjective homomorphism) Frobenius endomorphism Monomorphism (injective homomorphism) Lang. Algebra. p. 10. Lang. Algebra. p. 54. Jacobson (2009)
Endomorphism
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
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
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
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
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
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
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)
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
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
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
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
{\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
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
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
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
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
non-injective computations within a reversible framework. The core idea is to transform a non-injective function f: X → Y into a related injective function
Program Inversion, Interpretation, and Injectivization
Program_Inversion,_Interpretation,_and_Injectivization
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
sticking out of the wall for as long as possible while dealing with various objects being thrown at them, with the last remaining participant being the winner
List_of_Kickin'_It_episodes
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
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)
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
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
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
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
Category whose objects are measurable spaces and whose morphisms are measurable maps
the intersection of sigma-algebras. The monomorphisms in Meas are the injective measurable maps, the epimorphisms are the surjective measurable maps,
Category_of_measurable_spaces
"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
Eyes not aligning when looking at something
not properly align with each other when looking at an object. The eye that is pointed at an object can alternate. The condition may be present occasionally
Strabismus
Ratio of inertial to viscous forces acting on a liquid
although the effect is substantially more pronounced in gases. For a (solid) object in a fluid, the characteristic length will vary with temperature - solids
Reynolds_number
NASA satellite of the Explorer program
it is classified as a Super-Earth. As of September 2019, over 1000 TESS Objects of Interest (ToI) have been listed in the public database, at least 29
Transiting Exoplanet Survey Satellite
Transiting_Exoplanet_Survey_Satellite
Common rootkit technique
Direct kernel object manipulation (DKOM) is a common rootkit technique for Microsoft Windows to hide potentially damaging third-party processes, drivers
Direct kernel object manipulation
Direct_kernel_object_manipulation
Viral infection of sheep and goats
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)
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