My professor is introducing Triangulated Categories, and the examples given so far are:
- K(R) - the homotopy category of R-module complexes
- D(R)- the derived category of R-module complexes
- $\underline{\text{R-Mod}}$ - the stable category of R-modules
- HMF($S , x$) - the homotopy category of matrix factorizations
- Various triangulated subcategories of these as well
My question is this: are there examples of non-trivial triangulated categories that are not non-trivial quotient categories? If there are any, are they useful?
NOTE: By "quotient categories", I mean categories that one constructs by taking quotienting out an equivalence relation on morphisms and non-trivial quotient categories are those which are categories that are quotiented with themselves.