Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | Last revision Both sides next revision | ||
|
my_nice_nats [2014/11/02 13:46] nikolaj |
my_nice_nats [2014/11/03 16:58] nikolaj |
||
|---|---|---|---|
| Line 8: | Line 8: | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | That silly name... I made it up. | + | That silly name ... I made it up. |
| - | The point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats. | + | 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 ${\bf C}$. |
| - | === In words === | + | 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]]. |
| - | Here we have a situation with a functor $G$ from a ${\bf C}$, which is tame enough so that after mapping back to ${\bf C}$ via $F$, the "deforming" effect can be repaired by a natural transformation $\alpha:\mathrm{nat}(FG,1_{\bf C})$. | + | |
| === Theorems === | === Theorems === | ||
| - | Since $FG$ is a functor and there is a natural transformation, the structural properties "around" $FGX$ and $X$ are equivalent. However, only when the nats are isomorphisms (as in [[my equivalence of categories]]) is $F$ fully faithful and dense. | + | Only when the nats are isomorphisms (as in [[my equivalence of categories]]) is $F$ fully faithful and dense. |
| ==== Parents ==== | ==== Parents ==== | ||
