Get to know Adjoint better with 10+ real example sentences, the meaning.
Adjoint meaning
- Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugation (either inner automorphism or complex conjugation).
- That is related to another functor by an adjunction.
- Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
Using Adjoint
- The main meaning on this page is: Used in certain contexts, in each case involving a pair of transformations, one of which is, or is analogous to, conjugation (either inner automorphism or complex conjugation). | That is related to another functor by an adjunction. | Having a relationship of the nature of an adjoint (adjoint curve); sharing multiple points with.
- In the example corpus, adjoint often appears in combinations such as: left adjoint, adjoint to, adjoint functors.
Context around Adjoint
- Average sentence length in these examples: 24.4 words
- Position in the sentence: 8 start, 7 middle, 5 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis for Adjoint
- In this selection, "adjoint" usually appears near the start of the sentence. The average example has 24.4 words, and this corpus slice is mostly made up of statements.
- Around the word, left, self, functors, functor and equivalence stand out and add context to how "adjoint" is used.
- Recognizable usage signals include is left adjoint to g and a left adjoint. That gives this page its own corpus information beyond isolated example sentences.
- By corpus frequency, "adjoint" sits close to words such as abakaliki, abbasi and abductors, which helps place it inside the broader word index.
Example types with adjoint
The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:
Adjoint functors arise everywhere, in all areas of mathematics. (9 words)
Adjoint property is discussed in more general context in Hofman & Morris (2007) (e. (13 words)
A particular case of this happens when a continuous functor admits a left adjoint. (14 words)
Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups. (42 words)
Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. (39 words)
The equivalent symmetric definitions involving adjunctions and the symmetric language of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage of making this fact explicit. (38 words)
Example sentences (20)
An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous.
The equivalent symmetric definitions involving adjunctions and the symmetric language of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage of making this fact explicit.
We say that F is left adjoint to G, and G is right adjoint to F. Note that G may have itself a right adjoint that is quite different from F (see below for an example).
Conversely, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′.
Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.
Let F and G be a pair of adjoint functors with unit η and co-unit ε (see the article on adjoint functors for the definitions).
This article provides several such definitions: * The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint.
This means that T is left adjoint to the forgetful functor U (see the section below on relation to adjoint functors).
Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.
Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups.
Adjoint functors arise everywhere, in all areas of mathematics.
Adjoint property is discussed in more general context in Hofman & Morris (2007) (e.
Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint.
Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.
A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection).
A particular case of this happens when a continuous functor admits a left adjoint.
A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.
Dually, if G is additive with a left adjoint F, then F is also additive.
Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint ) linear operator acting on the state space.
For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits.
Common combinations with adjoint
These word pairs occur most frequently in English texts:
- left adjoint 20×
- adjoint to 16×
- adjoint functors 13×
- right adjoint 10×
- adjoint functor 8×
- of adjoint 7×
- an adjoint 6×
- to adjoint 2×
- adjoint then 2×
- adjoint of 2×