# Differences

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 my_nice_nats [2014/11/03 16:58]nikolaj my_nice_nats [2014/12/04 16:29] (current)nikolaj Both sides previous revision Previous revision 2014/12/04 16:29 nikolaj 2014/11/03 16:58 nikolaj 2014/11/02 13:46 nikolaj 2014/11/02 13:45 nikolaj 2014/11/01 18:16 nikolaj 2014/10/31 20:05 nikolaj 2014/10/31 20:00 nikolaj 2014/10/31 19:59 nikolaj 2014/10/31 19:58 nikolaj 2014/10/31 19:57 nikolaj 2014/10/31 19:43 nikolaj created2014/10/31 19:43 nikolaj removed2014/10/31 19:42 nikolaj created 2014/12/04 16:29 nikolaj 2014/11/03 16:58 nikolaj 2014/11/02 13:46 nikolaj 2014/11/02 13:45 nikolaj 2014/11/01 18:16 nikolaj 2014/10/31 20:05 nikolaj 2014/10/31 20:00 nikolaj 2014/10/31 19:59 nikolaj 2014/10/31 19:58 nikolaj 2014/10/31 19:57 nikolaj 2014/10/31 19:43 nikolaj created2014/10/31 19:43 nikolaj removed2014/10/31 19:42 nikolaj created Line 4: Line 4: | @#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]].