This rolls connectedness and abstractness into the one condition, as follows. It had therefore better have some independently justifiable redeeming social value before burdening math dictionaries with its own name.
A category is a ring with many objects but no additive structure, and a presheaf is then a left module. A representable presheaf is the ring considered as a left module over itself (but there are lots of them because of the many objects), and a sieve - a subpresheaf of a representable presheaf - is a left ideal. The category is also a - just one - bimodule (or profunctor) over itself just as a ring is, and the concept under discusion, a subbimodule, is an ideal exactly as Max Kelly said (a 2-sided ideal, or 2-sided sieve). That's a justification by analogy with ring theory, though there is a gap: ideals of rings are good not just because they are subbimodules. They are also kernels of quotients, and once the additive structure is dropped then it is no longer true that quotients are equivalent to subbimodules. Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++