categories: Re: Sheaves as a localisation of separated presheaves

[Richard Garner <richard.garner@mq.edu.au>]

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













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