This is an old revision of the document!
My nice nats
Collection
| context | $F$ in ${\bf D}\longrightarrow{\bf C}$ |
| context | $G$ in ${\bf C}\longrightarrow{\bf D}$ |
| definiendum | $\langle\alpha,\beta\rangle$ in it |
| inclusion | $\alpha:\mathrm{nat}(FG,1_{\bf C})$ |
| inclusion | $\beta:\mathrm{nat}(1_{\bf D},GF)$ |
Discussion
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.
In words
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
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.
Parents
Context
Requirements
