Notes for “Class forcing and topos theory”

A few years back I gave a talk at the IHES (video), for the conference Topos à l’IHÉS. After much procrastination, I have finally dragged my notes for that talk into a reasonable form.

David Michael Roberts, Class forcing and topos theory, talk notes (2018) doi:10.4225/55/5b2252e3092af

Abstract: It is well-known that forcing over a model of material set theory corresponds to taking sheaves over a small site (a poset, a complete Boolean algebra, and so on). One phenomenon that occurs is that given a small site, all new subsets created are smaller than a fixed bound depending on the size of the site. There is a more general notion of forcing invented by Easton to create new subsets of arbitrarily large sets, namely class forcing, where one starts with a partially ordered class. The existing theory of class forcing is entirely classical, with no corresponding intuitionist theory as in ordinary forcing. Our understanding of its relation to topos theory is in its infancy, [[ but it is clear that class forcing is about taking small sheaves on a large site, or rather, considering colimits of large diagrams of sheaf toposes and their inverse image functors. ]](Added April 2017: this is incorrect! Jensen forcing gives new sets(=sheaves) which aren’t set-generic(=small). My thanks to Joel David Hamkins for patiently explaining this to me.) That these do not automatically form a topos means that the theory has interesting twists and turns. This talk will outline the theory of class forcing from a category/topos point of view, give examples and constructions, and finally a list of open questions — not least being whether an intuitionistic version of Easton’s theorem on the continuum function holds.

Note that these are really very much very slightly polished talk notes as I understood the topic at the time, and so they are no reflection of a finished paper. I also had to retract a statement as noted in the abstract above, although this was merely a sweeping statement about the scope of the results, none of the actual results were in error. Anyway, it’s better to have these notes out and about. Comments and questions welcome…

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.