What is this quasi-coherence condition? What is its place among the general concepts of homological algebra known to the contemporary algebraists? It is the condition of left Ext^{0,1}-perpendicularity to a certain set of objects of injective dimension 1, as in the paper of Geigle and Lenzing. The quasi-coherent sheaves are the left perpendicular class to a set of objects of injective dimension 1 in the category of modules over a big ring.

According to yours truly, contraherent cosheaves on a scheme are representations of a certain quiver with relations ( = modules over a certain big ring) satisfying the additional "contraherence" and "contraadjustness" conditions. What are these contraherence and contraadjustness conditions?

These are the conditions of right perpendicularity to a certain set of objects of projective dimension 2. The contraherent cosheaves are the right Ext^{0,1,2}-perpendicular class to a set of objects of projective dimension 2 in the category of modules over a big ring.

That is why the quasi-coherent sheaves are an abelian category, while the contraherent cosheaves are an exact category. The left/right perpendicular class to a set/class of objects of injective/projective dimension 1 in an abelian category is an abelian category (as Geigle and Lenzing already observed), while the Ext^*-perpendicular class to a set/class of objects of homological dimension more than 1 in an abelian category is an exact category (since it is a full subcategory closed under extensions).