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adjunction
adjunction (əˈdʒʌŋkʃən) [ad. L. adjunctiōn-em, n. of action, f. adjunct- ppl. stem of adjung-ĕre: see adjunct. Cf. Fr. adjonction (14th c. in Littré.)] 1. The joining on or adding of a thing or person (to another).1618 Raleigh Rem. (1644) 270 That supposition, that your Majesties Subjects give nothi... Oxford English Dictionary
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Adjunction formula
Inversion of adjunction The adjunction formula is false when the conormal exact sequence is not a short exact sequence. Hence which agrees with the adjunction formula. wikipedia.org
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Using "adjunction" to refer to the act of taking adjoints of operators I have an especially flabby terminology question. > How acceptable is it, in your opinion, to use the word "adjunction" to refer to the process ...
Sure why not, right? Who's it going to hurt?
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Adjunction space
Adjunction spaces are also used to define connected sums of manifolds. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes. J.H.C. wikipedia.org
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Adjunction space $$X\cup_f Y = (X\amalg Y)/\\{f(A)\sim A\\}$$ This is the definition of adjunction space in Wikipedia. I wasn't able to understand some signs: $/$ and $\amalg$. What do they mean? Thanks.
The question is solved as in the comments.
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Quillen adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories This adjunction (LF, RG) is called the derived adjunction. wikipedia.org
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Axiom of adjunction
The adjunction operation is also used as one of the operations of primitive recursive set functions. In fact, empty set and adjunction alone (without extensionality) suffice to interpret . (They are mutually interpretable.) wikipedia.org
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Adjunction with reversed elements There is an adjunction between $L$ and $R$ when: $$ \text{Hom}(LA,B) \approx \text{Hom}(A,RB) $$ Is there something related we can say when instead we have: $$ \text{Hom}(B,LA) \ap...
A **contravariant adjunction on the right** consists of _contravariant_ functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C} Note that we can equally well think of this as an ordinary (or covariant) adjunction $$\mathcal{D}^\mathrm{op} (F X, Y) \cong \mathcal{C} (X, G Y)$$ where
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Monoidal adjunction
A monoidal adjunction between two lax monoidal functors and is an adjunction between the underlying functors, such that the natural transformations This adjunction lifts to a monoidal adjunction ⊣ if and only if the lax monoidal functor is strong. wikipedia.org
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What relationship do two adjunctions imply? Suppose we have three categories $A,B,C$, and we have an adjunction between $A$ and $C$ and we also have another adjunction bewteen $B$ and $C$. What relationship does this ...
That certainly doesn't imply $A$ and $B$ are equivalent, and doesn't even imply there exists any adjunction between $A$ and $B$.
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Tensor-hom adjunction
In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair: This is made more precise below. Counit and unit Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. wikipedia.org
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Isomorphism through adjunction An adjunction $F \dashv G$ gives a morphism $\phi(f) : A \to G B$ to each morphism $f : F A \to B$. Does $\phi(f)$ have any special property if I know that $f : F A \to B$ is an isomorph...
The only special properties it will have are those that the unit has, because the transpose of $\mathrm{id} : F A \to F A$ is precisely the unit $\eta_A : A \to G F A$. So, for example, the triangle identities imply that $\eta_{G A} : G A \to G F G A$ and $F \eta_A : F A \to F G F A$ are split monom...
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Adjunction Space , Collapsing boundary to a point I was studying Adjunction space from some lecture notes. I have the following Question, Suppose I take $X=D^{2}$, closed unit disk in plane. I take $A=S^{1}$ , the bo...
By definition of the adjunction space, if $Y$ is a one-point space then $X \sqcup_f Y = X/A$ .
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Is the unit of an adjunction an epimorphism? I was wondering, does the unit of an adjunction of functors need to be an epimorphism in general?
A common example of an adjunction is the free functor and forgetful functor for a kind of universal algebra -- e.g. real vector spaces. The unit of said adjunction is the map that sends a set $S$ to the (underlying set of the) vector space whose basis is $S$.
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Is the skeleton-coskeleton adjunction $sSet$-enriched? Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-...
Well, first we would have to have simplicially enriched functors. But $\mathrm{sk}_n : \mathbf{sSet} \to \mathbf{sSet}$ is _not_ simplicially enriched. Indeed, any simplicially enriched functor must preserve simplicial homotopy equivalences, but $\mathrm{sk}_n$ does _not_ preserve simplicial homotop...
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