why the concept of compactly (or well) generated in triangulated categories is introduced?



 why the concept of compactly (or well) generated in triangulated categories is introduced?. 
Compactly generated ones are the biggest class of triangulated categories where the [Brown representability theorem](http://en.wikipedia.org/wiki/Brown%27s_representability_theorem) holds and can be checked using the same proof as for the stable homotopy category. This class of triangulated categories is not closed under standard operations, such as Verdier quotients, localizations... Well generated categories contain compactly generated ones, are closed under this kind of operations, and satisfy Brown representability. See Neeman's book for further details.


