categories: Sheaves as a localisation of separated presheaves

categories: Sheaves as a localisation of separated presheaves

[Richard Garner]

Dear 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.

The proof is rather easy but I am wondering if there is the even easier
possibility of citing something.

Thanks!

Richard



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