Differences

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

Link to this comparison view

Both sides previous revision Previous revision
my_nice_nats [2014/11/03 16:58]
nikolaj
my_nice_nats [2014/12/04 16:29] (current)
nikolaj
Line 4: Line 4:
 | @#55CCEE: context ​    | @#55CCEE: $G$ in ${\bf C}\longrightarrow{\bf D}$ | | @#55CCEE: context ​    | @#55CCEE: $G$ in ${\bf C}\longrightarrow{\bf D}$ |
 | @#FFBB00: definiendum | @#FFBB00: $\langle\alpha,​\beta\rangle$ in it | | @#FFBB00: definiendum | @#FFBB00: $\langle\alpha,​\beta\rangle$ in it |
-| @#AAFFAA: inclusion ​  | @#AAFFAA: $\alpha:\mathrm{nat}(FG,1_{\bf C})$ | +| @#AAFFAA: inclusion ​  | @#AAFFAA: $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ | 
-| @#AAFFAA: inclusion ​  | @#AAFFAA: $\beta:\mathrm{nat}(1_{\bf D},GF)$ |+| @#AAFFAA: inclusion ​  | @#AAFFAA: $\beta:​1_{\bf D}\xrightarrow{\bullet}GF$ |
  
 ==== Discussion ==== ==== Discussion ====
 That silly name ... I made it up.  That silly name ... I made it up. 
  
-The natural transformation $\beta:\mathrm{nat}(1_{\bf D},GF)$ squeezes every set $X\in {\bf D}$ into a set $GFX\in {\bf D}$ (although this need not be surjective or injective). The natural transformation $\alpha:\mathrm{nat}(FG,1_{\bf C})$ squeezes all sets $FGX$ in the image of $FG$ back into $X$. The latter operation gets rid of lots $FG$'s without changing the structural properties of ${\bf C}$.+The natural transformation $\beta:​1_{\bf D}\xrightarrow{\bullet}GF$ squeezes every set $X\in {\bf D}$ into a set $GFX\in {\bf D}$ (although this need not be surjective or injective). The natural transformation $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$ squeezes all sets $FGX$ in the image of $FG$ back into $X$. The latter operation gets rid of lots $FG$'s without changing the structural properties of ${\bf C}$.
  
 The point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats. In the former case, the two nats actually shift the whole content of a category internally. In the latter case, the two nats end up defining the shifting operations of a [[monad]]. The point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats. In the former case, the two nats actually shift the whole content of a category internally. In the latter case, the two nats end up defining the shifting operations of a [[monad]].
Link to graph
Log In
Improvements of the human condition