Highest weight approximations of chain abelian categories

I will describe the construction of highest weight categories as abelian envelopes of thin exact categories, i.e. exact categories with 'full exceptional collections'. I will discuss a source of examples of thin exact categories given by slender filtrations on abelian categories, i.e. filtrations with graded factors equivalent to the category of vector spaces. In geometry, slender filtrations appear naturally for sandwiched surface singularities. For cyclic quotient singularities, or their algebraic counterpart - chain abelian categories, I will classify all possible slender filtrations and calculate the abelian envelopes of the thin exact categories they give. I will show that they are all derived equivalent. This is a joint work in progress with A. Bondal.