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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 21st 2018

    I added the suggestion that there is a version for HoTT.

    diff, v14, current

    • CommentRowNumber2.
    • CommentAuthorThomas Holder
    • CommentTimeJun 29th 2020

    Added references to

    • Anders Kock, Universal Projective Geometry via Topos Theory , JPAA 9 (1976) pp.1-24.

    • Anders Kock, Elementwise semantics in categories with pull-backs , arXiv:2004.14731 (2020). (abstract)

    diff, v20, current

    • CommentRowNumber3.
    • CommentAuthorjulia9367
    • CommentTimeApr 17th 2022
    • (edited Apr 17th 2022)

    I think it would be nice to add the specialisation of sheaf semantics for any topos of sheafs of sets on a site. In this case we have nice concrete interpretations of the logic. E.g.:

    • it suffices to consider generalized elements of the form x(U)x \in \mathcal{F}(U)
    • any formula is true if it is true on an open cover
    • \implies = implies on all open subsets
    • ¬\neg = false on all open subsets
    • ¬¬\neg\neg = false that (it is false on all open subsets) of all open subsets = true on a dense open subset

    [1] is a beautiful application of this to scheme theory simplifying many statements and proofs.

    [1]: Using the internal language of toposes in algebraic geometry, Ingo Blechschmidt, https://arxiv.org/abs/2111.03685

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