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## Definition of the noun adjunction

What does adjunction mean as a name of something?

1. an act of joining or adjoining things

### Alternative definition of the noun adjunction

noun

1. [category theory] A form of similarity between a pair of categories $\mathcal\left\{C\right\}$ and $\mathcal\left\{D\right\}$ which is weaker than equivalence, which in turn is weaker than isomorphism. Given functors $F:\mathcal\left\{C\right\} \rightarrow \mathcal\left\{D\right\}$ and $G:\mathcal\left\{D\right\} \rightarrow \mathcal\left\{C\right\}$, F is "left adjoint" of G, and G "right adjoint" of F, denoted as $F \dashv G$, if
2. there are a pair of natural transformations $\eta: \mbox\left\{id\right\}_\mathcal\left\{C\right\} \rightarrow GF$ and $\epsilon: FG \rightarrow \mbox\left\{id\right\}_\mathcal\left\{D\right\}$ satisfying the following "triangle identities":
3. # $F \eta : F \rightarrow FGF$ composed with $\epsilon G: FGF \rightarrow F$ commutes with $\mbox\left\{id\right\}_F: F \rightarrow F$ and
4. # $\eta G: G \rightarrow GFG$ composed with $F \epsilon: GFG \rightarrow G$ commutes with $\mbox\left\{id\right\}_G: G \rightarrow G$.
5. there is a natural isomorphism $\alpha: \mathcal\left\{D\right\}\left(FX,Y\right) \cong \mathcal\left\{C\right\}\left(X,GY\right)$, which is natural in the sense of being "natural in X and Y", where
6. # "natural in X" means that for every $f:X\rightarrow X\text{'}$, $\alpha:\mathcal\left\{D\right\}\left(FX\text{'},Y\right) \rightarrow \mathcal\left\{C\right\}\left(X\text{'},GY\right)$ composed with $\mathcal\left\{C\right\}\left(f,GY\right):\mathcal\left\{C\right\}\left(X\text{'},GY\right) \rightarrow \mathcal\left\{C\right\}\left(X,GY\right)$ commutes with $\mathcal\left\{D\right\}\left(Ff,Y\right):\mathcal\left\{D\right\}\left(FX\text{'},Y\right) \rightarrow \mathcal\left\{D\right\}\left(FX,Y\right)$ composed with $\alpha:\mathcal\left\{D\right\}\left(FX,Y\right)\rightarrow \mathcal\left\{C\right\}\left(X,GY\right)$;
7. # "natural in Y" means that for every $f:Y \rightarrow Y\text{'}$, $\alpha:\mathcal\left\{D\right\}\left(FX,Y\right) \rightarrow \mathcal\left\{C\right\}\left(X,GY\right)$ composed with $\mathcal\left\{C\right\}\left(X,Gf\right):\mathcal\left\{C\right\}\left(X,GY\right) \rightarrow \mathcal\left\{C\right\}\left(X,GY\text{'}\right)$ commutes with $\mathcal\left\{D\right\}\left(FX,f\right):\mathcal\left\{D\right\}\left(FX,Y\right) \rightarrow \mathcal\left\{D\right\}\left(FX,Y\text{'}\right)$ composed with $\alpha:\mathcal\left\{D\right\}\left(FX,Y\text{'}\right) \rightarrow \mathcal\left\{C\right\}\left(X,GY\text{'}\right)$.

## Explanation

Adjunction: In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.

## Printed dictionaries and other books with definitions for Adjunction

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### Rings, Modules, and Algebras in Stable Homotopy Theory (2007)

by Anthony D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May

We claim that if if M is E-local, then the counit s : mono M → M of the first adjunction is a weak equivalence of R-modules. To see this, consider the commutative diagrams: ØRe(RE AR So, m*M) ØR(n” (RE AR S4), m*moM) § 2R ( S3, m*mżM) ...

### The Mathematics of Language (2011)

12th Biennial Conference, MOL 12, Nara, Japan, September 6-8, 2011, Proceedings by Makoto Kanazawa, Marcus Kracht, Hiroyuki Seki, András Kornai

Adjunction is an operation on a derivation tree T where a tree αfrom A is adjoined at a node e ∈ T ifthe root ofα shares the same symbol as e. Ifso, then the subtree rooted at e in T is deletedfrom T ...

### Exact Categories and Categories of Sheaves (2006)

by M. Barr, P. A. Grillet, D. H. van Osdol

The inner adjunction is the map X in which x ...

### Natural Biodynamics (2005)

by Vladimir G. Ivancevic, Tijana T. Ivancevic

The simplest form of this adjunction is the basis of all neural activity, namely, the reflex sensory-motor action.

### International Encyclopedia of Linguistics (2003)

See Adjunction; Phrase Structure; and Clause, article on Complementation. ADJUNCTION. In the theory of generative grammar, transformations are said to map phrase markers into other phrase markers. Transformational operations are ...

### A Dictionary of Literary Devices (1991)

Gradus, A-Z by Bernard Marie Dupriez

Adynaton Adjunction may occur at the beginning (protozeugma), in the middle ( mesozeugma), or at the end of a sentence (hypozeugma). R2: Nowadays adjunction is a mode of syntactic development, especially in spoken language. Indeed ...

### The Concise Oxford Dictionary of Linguistics (2007)

by P.H. Matthews

adjunction 1. Any operation, especially in syntax, by which an element is added next to another.

### Advances in Generative Lexicon Theory (2012)

by James Pustejovsky, Pierrette Bouillon, Hitoshi Isahara, Kyoko Kanzaki, Chungmin Lee

Let us first consider a regular non-collapsing adjunction process. For two clauses CL1 , whose qualia structure is QS 1 , and CL2, whose qualia structure is QS2, suppose CL1 is adjoined to CL2. The question is what happens to QS1 and QS2 ...

### A Student's Dictionary of Language and Linguistics (2014)

adjunction In transformational grammar,any procedure for inserting material ...

### Encyclopaedia of Mathematics (2012)

Supplement by Michiel Hazewinkel

ADJUNCTION THEORY det Tx., where T: is the cotangent bundle of X. For simplicity, below the objects are considered over the complex numbers. The book [1] is a general reference with full coverage of the literature for the whole theory with ...

### Encyclopedia of Rhetoric (2001)

by Thomas O. Sloane

adjunction (the addition of a constituent to the right or left of some other constituent); (3) substitution (a combination of deletion ...

### Routledge Dictionary of Language and Linguistics (2006)

Chomsky adjunction 73 classifying verb) in the southern language Itonama. One unusual syntactic trait: the subject of past tense sentences is marked with the genitive. Word order usually ...

### Comprehensive Dictionary of Electrical Engineering, Second Edition (2005)

by Philip A. Laplante

Adjunction duality: The dilation and erosion by B, namely δB : X→ X ⊕ B and ε B : X→ X ⊖ B, form an adjunction, which means that for every X, Y we have δB ( X) ≤ Y ⇐⇒ X ≤ ε B (Y), the dilation of X is below Y if and only if X us below the ...

### Longman Dictionary of Language Teaching and Applied Linguistics (2013)

by Jack C. Richards, Richard W. Schmidt

adjunction e they can be used in a comparative or superlative form, e.g. the bag seems heavier now. see also complement, comparative, attributive adjective adjective complement n see complement adjective phrase n a phrase that functions ...

## Online dictionaries and encyclopedias with entries for Adjunction

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Video shows what adjunction means. The act of joining, the thing joined or added.. A form of similarity between a pair of categories mapped to each other by ...

## Scrabble value of A1D2J8U1N1C3T1I1O1N1

The value of this 10-letter word is 20 points, but it's not an accepted word in the Official Scrabble Players Dictionary.

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