well-generated triangulated category
A well-generated triangulated category is a strengthening of the notion of compactly generated triangulated category which was introduced by Neeman, 2001. The following definition is from (Krause) and is somewhat shorter and more natural than Neeman’s original definition.
Let be a triangulated category with arbitrary coproducts. Then is well-generated in the sense of Neeman if and only if there exists a set of objects satisfying:
an object of is zero if for all ;
for every set of maps in , the induced map is surjective for all whenever is surjective for all and all .
the objects of are -small for some cardinal .
We recall that to say an object is -small in a triangulated category is to say that every map factors through some whenever .
- Henning Krause, On Neeman’s Well Generated Triangulated Categories, Documenta Mathenatica 6 (2001) (pdf).
- Amnon Neeman, Triangulated Categories, Annals of Mathematics Studies 148, Princeton University Press (2001).