The notion of epipresheaf is formally dual to the more standard notion of separated presheaf: where the latter has a monomorphism, the former has an epimorphism.
From this perspective a sheaf is a presheaf satisfying two properties: the epipresheaf condition and the “monopresheaf” (or separated presheaf) condition. Thus there are epipresheaves, monopresheaves and sheaves.
A presheaf A is called epipresheaf if for any local isomorphism f:X→Y the map A(Y)→A(X) is an epimorphism
The notion is introduced in