[FOM] Reference request for category-theoretic presentation of forcing

[Christian Espindola]

Chapter VI of the book "Sheaves in geometry and logic" by MacLane and
Moerdijk might be what you are looking for. It contains a topos-theoretic
presentation of forcing, more precisely, two toposes of sheaves with the
double-negation topology that show the consistency of ~CH and ~AC: the so
called Cohen topos (a Boolean topos satisfying the axiom of choice where
the continuum hypothesis fails) and a Boolean topos found by Freyd where
the axiom of choice fails.

The relation to the method of forcing becomes clear when considering the
Kripke-Joyal semantics, explained in the same chapter, as well as the
observation that models of ZF can be obtained from Boolean Grothendieck
toposes by mimicking the construction of the cumulative hierarchy. That
said, the mathematical content of these category-theoretic proofs are
essentially the same as the forcing technique via Boolean-valued models,
and any other background that you might need is available in that same book.

Finally, chapter VI also contains a topos-theoretic axiomatization of the
category of sets and the proof of equiconsistency with a weak version of Z.

Best,

Christian

On Mon, Jul 3, 2017 at 1:44 PM, Neil Barton <bartonna at gmail.com> wrote:

> Dear All,
>
> A short reference request: I'm interested in the category-theoretic
> presentations of set-theoretic forcing (e.g. showing that ~CH is consistent
> with ZFC).
>
> As someone with a reasonable knowledge of set theory (inner models and the
> forcing construction are certainly fine) and a basic knowledge of topos
> theory (subobject classifiers, algebras of subobjects, sheaves etc.) what's
> the best reference here? Would that be the Appendix to Bell's *Boolean-Valued
> Models and Independence Proofs*, or are there other references? I would
> like a little more detail on the wider implications of this way of cashing
> out the results, in particular how they relate to category theory/set
> theory more generally.
>
> Best Wishes,
>
> Neil
>
> --
> Dr. Neil Barton
> Postdoctoral Research Fellow
> Kurt Gödel Research Center for Mathematical Logic
> University of Vienna
>
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