categories: Re: Sheaves as a localisation of separated presheaves

categories: Re: Sheaves as a localisation of separated presheaves

[Richard Garner]

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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categories: Re: Sheaves as a localisation of separated presheaves

[Richard Garner]

> the localisation functor Psh(C) --> Psh(C)[D^-1], which obviously
> preserves colimits,

Here it is perhaps better to say "which does not obviously preserve
colimits". But the point remains the same: Psh(C) --> Psh(C)[D^-1] is
localisation in the wrong 2-category, CAT rather than COCTS.

Richard



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categories: Re: Sheaves as a localisation of separated presheaves

[Vaughan Pratt]

" I realise I am basically talking to myself at this point but figure I
should try to set the record straight!"

Me too.  But what I love are the ever so sparse instances of evidence to
the contrary that come from unexpected quarters...  ;)

A big part of the problem is that there are fewer concepts than there are
words for them.  Unless you know every word for any given concept, you may
not realize that someone is agreeing with you.

Vaughan


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