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hom-set_adjunction [2016/02/11 13:15] nikolaj |
hom-set_adjunction [2016/02/11 13:23] nikolaj |
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== Idea == | == Idea == | ||
- | More generally, view the left adjoint $F$ as A-"thickening" of ist argument ($X$) and view $G$ as the A-indexing of aspects of it's argument $Y$. | + | More generally, |
+ | view the left adjoint $F$ as A-"thickening" of ist argument ($X$) and view $G$ as the A-indexing of aspects of it's argument $Y$. | ||
- | == Example from Algebra == | + | If ${\bf C}\neq{\bf D}$, then viewing $G$ as indexing may be harder. |
- | + | ||
- | For example in the category of groups | + | |
- | + | ||
- | $\mathrm{Hom}(X\otimes A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y))$ | + | |
== Currying == | == Currying == | ||
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Here the A-"thickening" side says you have more arugments to prove $Y$ to begin with, while the $A$-"indexing" side means you only demonstrate A-conditional truth of $Y$. | Here the A-"thickening" side says you have more arugments to prove $Y$ to begin with, while the $A$-"indexing" side means you only demonstrate A-conditional truth of $Y$. | ||
+ | |||
+ | == Example from Algebra == | ||
+ | |||
+ | For example in the category of groups | ||
+ | |||
+ | $\mathrm{Hom}(X\otimes A,Y)\cong\mathrm{Hom}(X,\mathrm{Hom}(A,Y))$ | ||
== Galois connection == | == Galois connection == | ||
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$(A\times Y^A)$ to $Y$. | $(A\times Y^A)$ to $Y$. | ||
- | The first can only be | + | The first can only be a direct embedding |
$\eta_Y(y):=\lambda a.\, \langle a,y\rangle$ | $\eta_Y(y):=\lambda a.\, \langle a,y\rangle$ | ||
- | and the second | + | and the second is an evaluation |
$\epsilon_Y(\langle a,f\rangle) := f(a)$ | $\epsilon_Y(\langle a,f\rangle) := f(a)$ |