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my_nice_nats [2014/11/01 18:16] nikolaj |
my_nice_nats [2014/11/02 13:45] nikolaj |
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==== Discussion ==== | ==== Discussion ==== | ||
That silly name... I made it up. | 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 === | === In words === | ||
- | 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. | + | 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 $\alpha:\mathrm{nat}(FG,1_{\bf C})$. |
- | The point is that [[my equivalence of categories]] and [[Counit-unit adjunction]] are two different important special cases of nice nats. | + | === Theorems === |
+ | Since $FG$ is a functor and there is a natural transformation, the structural properties "around" $FGX$ and $X$ are equivalent. However, only when that nat is an isomorphisms is $F$ fully faithful and dense. | ||
==== Parents ==== | ==== Parents ==== |