# Adjunction

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## Definition of the noun adjunction

What does adjunction mean as a name of something?

**noun** - plural: adjunctions

- an act of joining or adjoining things
- lexical domain: Acts - nouns denoting acts or actions
- synonym of adjunction: junction
- more generic words: connection / connexion / joining = the act of bringing two things into contact

### Alternative definition of the noun adjunction

**noun**

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

## 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

Click on a title to look inside that book (if available):

### 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)

by Larry Trask

**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)

by Hadumod Bussmann

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 about Adjunction

**Adjunction Meaning**

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 A_{1}D_{2}J_{8}U_{1}N_{1}C_{3}T_{1}I_{1}O_{1}N_{1}

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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