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hom-set_adjunction [2016/02/11 13:16] 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 == | ||
| Line 46: | Line 43: | ||
| 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 == | ||
