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Adjunction

Definition of the noun adjunction

What does adjunction mean as a name of something?

noun - plural: adjunctions

  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{C} and \mathcal{D} which is weaker than equivalence, which in turn is weaker than isomorphism. Given functors F:\mathcal{C} \rightarrow \mathcal{D} and G:\mathcal{D} \rightarrow \mathcal{C} , 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{id}_\mathcal{C} \rightarrow GF and \epsilon: FG \rightarrow \mbox{id}_\mathcal{D} satisfying the following "triangle identities":
  3. # F \eta : F \rightarrow FGF composed with \epsilon G: FGF \rightarrow F commutes with \mbox{id}_F: F \rightarrow F and
  4. # \eta G: G \rightarrow GFG composed with F \epsilon: GFG \rightarrow G commutes with \mbox{id}_G: G \rightarrow G .
  5. there is a natural isomorphism \alpha: \mathcal{D}(FX,Y) \cong \mathcal{C}(X,GY) , 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', \alpha:\mathcal{D}(FX',Y) \rightarrow \mathcal{C}(X',GY) composed with \mathcal{C}(f,GY):\mathcal{C}(X',GY) \rightarrow \mathcal{C}(X,GY) commutes with \mathcal{D}(Ff,Y):\mathcal{D}(FX',Y) \rightarrow \mathcal{D}(FX,Y) composed with \alpha:\mathcal{D}(FX,Y)\rightarrow \mathcal{C}(X,GY);
  7. # "natural in Y" means that for every f:Y \rightarrow Y', \alpha:\mathcal{D}(FX,Y) \rightarrow \mathcal{C}(X,GY) composed with \mathcal{C}(X,Gf):\mathcal{C}(X,GY) \rightarrow \mathcal{C}(X,GY') commutes with \mathcal{D}(FX,f):\mathcal{D}(FX,Y) \rightarrow \mathcal{D}(FX,Y') composed with \alpha:\mathcal{D}(FX,Y') \rightarrow \mathcal{C}(X,GY').

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):

Google previewRings, 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) ...

Google previewThe 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 ...

Google previewExact 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 ...

Google previewNatural 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.

Google previewInternational 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 ...

Google previewA 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 ...

Google previewThe 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.

Google previewAdvances 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 ...

Google previewA Student's Dictionary of Language and Linguistics (2014)

by Larry Trask

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

Google previewEncyclopaedia 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 ...

Google previewEncyclopedia 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 ...

Google previewRoutledge 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 ...

Google previewComprehensive 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 ...

Google previewLongman 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 ...

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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 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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