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--- my_nice_nats [2014/11/01 18:16]
nikolaj
+++ my_nice_nats [2014/11/02 13:46]
nikolaj
@@ Line 9: / Line 9: @@
 ==== 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 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 the nats are isomorphisms (as in [[my equivalence of categories]]) is $F$ fully faithful and dense.
 ==== Parents ====