My nice nats


context $F$ in ${\bf D}\longrightarrow{\bf C}$
context $G$ in ${\bf C}\longrightarrow{\bf D}$
definiendum $\langle\alpha,\beta\rangle$ in it
inclusion $\alpha:FG\xrightarrow{\bullet}1_{\bf C}$
inclusion $\beta:1_{\bf D}\xrightarrow{\bullet}GF$


That silly name … I made it up.

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.


Only when the nats are isomorphisms (as in my equivalence of categories) is $F$ fully faithful and dense.




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