# Abelian envelopes of exact categories

I will introduce the left and right abelian envelopes of an exact category. They are abelian categories with a universal left or right exact functor from the original exact category. I will give two examples of abelian envelopes.

I will show that the highest weight categories are left/right abelian envelopes of exact categories generated by an analogue of a full exceptional collection. From this point of view, Ringel duality is the passage between the left and the right envelope.

I will also discuss an appropriate quotient of the category of coherent sheaves and show that it is the right abelian envelope of the category of reflexive sheaves on a Cohen Macaulay surface.

This is joint work with A. Bondal.