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my_nice_nats [2014/11/01 18:16]
nikolaj
my_nice_nats [2014/11/03 16:58]
nikolaj
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 ==== Discussion ==== ==== Discussion ====
-That silly name... I made it up. +That silly name ... I made it up. 
  
-=== In words === +The natural transformation $\beta:​\mathrm{nat}(1_{\bf D},GF)squeezes every set $X\in {\bf D}$ into 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 ${\bf C}$.
-Here we have a situation where there are two functorswhich don't really deform the category ​${\bf C}$ all that much: They are tame enough so that, by a natural transformation $\alpha:​\mathrm{nat}(FG,​1_{\bf C})$, their composite effect can be repaired ​back to unity.+
  
-The point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats.+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]]. 
 + 
 +=== Theorems === 
 +Only when the nats are isomorphisms (as in [[my equivalence of categories]]) is $F$ fully faithful and dense.
  
 ==== Parents ==== ==== Parents ====
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