I'm trying to learn about the topos of sheaves and the double negation topology to try to go through the independence of CH from a categorical perspective. I'm curious in general about what the standard set theoretic notions of forcing would look like in category theory.

For example: what corresponds to Prikry forcing in the topos of sheaves point of view? Is there a categorical interpretation of the Levy-Solovay theorem about large cardinals being unaffected by small forcing? Is this known or has it been investigated somewhere? Thanks in advance for any references or answers/comments.

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    $\begingroup$ Shouldn't the double-negation topology be called "the no-no topology"? :-) $\endgroup$
    – Asaf Karagila
    Jul 21 '20 at 7:45

The topos version of forcing with a poset $P$ regards $P$ as a category, forms the topos of presheaves on it, and then passes to the subtopos of double-negation sheaves. The presheaf topos amounts to strong forcing (which obeys only intuitionistic logic) and the sheaf subtopos amounts to the much more widely used weak forcing. This was, at least for the Cohen models violating CH, in the early work of Lawvere and Tierney, presented in Lawvere's paper "Quantifiers and sheaves" (Proceedings of the 1970 ICM in Nice. Big pdf (see p329), individual article). If I remember correctly, Marta Bunge worked out the topos version of the forcing that adjoins a Suslin tree (JPAA 1974, doi:10.1016/0022-4049(74)90020-6, see also the erratum).

More information about these topics is in work of John Bell (connecting it with Boolean-valued models, if I remember correctly) and in Peter Freyd's paper "All topoi are localic" (JPAA 1987, doi:10.1016/0022-4049(87)90042-9). Andre Scedrov and I included much of this material in the early, background sections of "Freyd's models for the axiom of choice" (Memoirs A.M.S. 404 (1989) --- this was before "404" became a widely known synonym for "file not found").

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    $\begingroup$ Thank you Andreas for your answer. Is there anything known about what Prikry's notion of forcing looks like as a category? More specifically: is there a suitable definition of measurable cardinal and singularizing sequence in category theory? $\endgroup$ Jul 20 '20 at 17:12
  • $\begingroup$ I would add the "subtopos of double-negation sheaves" can be described directly as sheaves on the poset taken as a site with the dense topology. $\endgroup$ Jul 21 '20 at 1:11
  • $\begingroup$ @DavidRoberts Right. Where topos folks say "double-negation sheaves", set folks say "every extension has an extension" (as if it were a single word because we've said it so often). $\endgroup$ Jul 21 '20 at 1:13
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    $\begingroup$ Here's Tierney's paper on his joint work with Lawvere: Tierney M. (1972) Sheaf theory and the continuum hypothesis. In: Lawvere F.W. (eds) Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics, vol 274https://doi.org/10.1007/BFb0073963. I think the mentioned work of Bell is all in Toposes and Local Set Theories, published by Dover, and with some lectures summarising the ideas published as Categories, toposes and sets. Synthese 51, 293–337 (1982). doi.org/10.1007/BF00485258, though there is a scanning error in this pdf that lost four pages. $\endgroup$ Jul 21 '20 at 1:40

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