Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
Last revision Both sides next revision
hom-set_adjunction [2016/02/11 13:22]
nikolaj
hom-set_adjunction [2016/04/07 14:19]
nikolaj
Line 15: Line 15:
 == An example in the category of sets == == An example in the category of sets ==
  
-Let both ${\bf C}$ and ${\bf D}$ be the category ${\bf Set}$, which has products and exponential objects. Fix some objects (sets) $A$ and $Y$. Many examples can be thought of as Variation ​of the pretty obvious relation+Let both ${\bf C}$ and ${\bf D}$ be the category ${\bf Set}$, which has products and exponential objects. Fix some objects (sets) $A$ and $Y$. Many examples can be thought of as variation ​of the pretty obvious relation
  
 $\mathrm{Hom}_{\bf Set}(*\times A,​Y)\cong\mathrm{Hom}_{\bf Set}(A,Y):= Y^A\cong\mathrm{Hom}_{\bf Set}(*,​Y^A)$ $\mathrm{Hom}_{\bf Set}(*\times A,​Y)\cong\mathrm{Hom}_{\bf Set}(A,Y):= Y^A\cong\mathrm{Hom}_{\bf Set}(*,​Y^A)$
Line 25: Line 25:
 $\mathrm{Hom}_{\bf Set}(X\times A,​Y)\cong\mathrm{Hom}_{\bf Set}(X,​Y^A)$ $\mathrm{Hom}_{\bf Set}(X\times A,​Y)\cong\mathrm{Hom}_{\bf Set}(X,​Y^A)$
  
-and this is hom-set adjunction ​+and this is hom-set adjunction ​
  
-$\mathrm{Hom}_{\bf Set}(FX,​Y)\cong\mathrm{Hom}_{\bf Set}(X,Y^A)$+$\mathrm{Hom}_{\bf Set}(FX,​Y)\cong\mathrm{Hom}_{\bf Set}(X,GY)$
  
 if we define the Action of $F$ on object via $FX:​=X\times A$ (Cartesian product) and let the action of $G$ on object be given by $GY:=Y^A$ (function space from $A$ to $Y$).  if we define the Action of $F$ on object via $FX:​=X\times A$ (Cartesian product) and let the action of $G$ on object be given by $GY:=Y^A$ (function space from $A$ to $Y$). 
Line 33: Line 33:
 == Idea == == Idea ==
 More generally, ​ 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 $Y$.+view the left adjoint $F$ as A-"​thickening"​ of ist argument ($X$) and view $G$ as the A-indexing'​s ​of aspects of it's argument $Y$.
  
 If ${\bf C}\neq{\bf D}$, then viewing $G$ as indexing may be harder. If ${\bf C}\neq{\bf D}$, then viewing $G$ as indexing may be harder.
- 
-== Example from Algebra == 
- 
-For example in the category of groups 
- 
-$\mathrm{Hom}(X\otimes A,​Y)\cong\mathrm{Hom}(X,​\mathrm{Hom}(A,​Y))$ 
  
 == Currying == == Currying ==
Line 48: Line 42:
 $\left((X\land A)\implies Y\right)\leftrightarrow\left(X\implies(A\implies Y)\right)$ $\left((X\land A)\implies Y\right)\leftrightarrow\left(X\implies(A\implies Y)\right)$
  
-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's" 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 ==
Link to graph
Log In
Improvements of the human condition