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my_nice_nats [2014/11/01 18:16] 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. |
| - | === In words === | + | 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 ${\bf C}$. |
| - | Here we have a situation where there are two functors, which 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 ==== | ||
