# Differences

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my_nice_nats [2014/11/03 16:58] nikolaj |
my_nice_nats [2014/12/04 16:29] (current) 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 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]]. |