Hi all,

If (C,J) is a site, then the category Sh(C) can be presented as a category of fractions Psh(C)[D^-1] where D is the class of J-dense monomorphisms. Morphisms in here are equivalence classes of partial maps X --> Y whose domain is dense in X, and where the equivalence relation is generated by the 2-cells of spans.

This is all well known. I am wondering if there is a reference for the following fact: if one restricts to separated presheaves, then every equivalence class of morphisms has a maximal representative, given by the dense partial maps X <--< X' ---> Y whose graph X' >---> X*Y is a J-closed monomorphism. So between separated presheaves, no quotienting is necessary---beyond that inherent in the notion of subobject---though now composition is no longer span composition on equivalence classes, but rather span composition followed by J-closure.

Looking more carefully, the proof of this seems to work fine if only Y is separated. So another way of saying the above is that the plus-construction on separated presheaves can be described without taking a quotient. However, thinking about this a bit more I should also fess up that my original statement is in error. Psh(C)[D^-1] is NOT the category of sheaves. It is the result of universally inverting the dense monos, but the category so obtained is not cocomplete. To get a cocomplete category, we need to invert the class of all bidense morphisms. How do we know Psh(C)[D^-1] isn't cocomplete? Well, if it were, then the localisation functor Psh(C) --> Psh(C)[D^-1], which obviously preserves colimits, would necessarily be sheafification Psh(C) --> Sh(C) (by the universal property of the latter), with as right adjoint the singular functor of C --> Psh(C) --> Psh(C)[D^-1]. But the monad induced by this adjunction on Psh(C) would then be the single plus construction, which is well-known not to be the reflector into the category of sheaves. (This had me worried for a while, as it seemed I had come up with an interesting proof of _|_.) On the other hand, it seems to be totally fine to describe Sh(C) as SepPsh(C)[D^-1], because the latter is the Kleisli category for the single-plus construction on SepPsh(C); and then the above quotient-free description does pertain. I realise I am basically talking to myself at this point but figure I should try to set the record straight! Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]