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my_nice_nats [2014/11/02 13:46] nikolaj |
my_nice_nats [2014/12/04 16:29] nikolaj |
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| @#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 point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats. | + | 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 ${\bf C}$. |
- | === 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 ==== |