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    • CommentRowNumber1.
    • CommentAuthorspitters
    • CommentTimeJul 7th 2018

    Relation to bunched logic

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    added to the Idea-section (here) a sentence pointing to doubly closed monoidal category.

    diff, v8, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    added (here) mentioning of the example of tangent \infty-toposes.

    It ought to be true that every \infty-topos arising from parameterized objects in an \otimes-monoidal Joyal locus carries the corresponding external \otimes-tensor product — but now that I say this I realize that this needs a formal argument which might be easy but I haven’t fully thought about.

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthormaxsnew
    • CommentTimeJan 15th 2024

    Observe that the overlap between Day convolution and slice over a monoid object agree.

    diff, v9, current

    • CommentRowNumber5.
    • CommentAuthormaxsnew
    • CommentTimeJan 15th 2024

    Also am I correct that the slice over a monoid object in a topos is always monoidal closed? Seems to follow by the AFT because the tensor product should preserve colimits in each argument because colimits are computed in the original category and the cartesian product preserves colimits in each argument.

    If so I’ll add that to the page too.

    • CommentRowNumber6.
    • CommentAuthorAlexanderCampbell
    • CommentTimeJan 16th 2024
    • (edited Jan 16th 2024)

    Yes, that’s true, and one can give an easy construction of the internal hom; see this MO answer.

    • CommentRowNumber7.
    • CommentAuthormaxsnew
    • CommentTimeJan 16th 2024

    I must be missing something, the construction in that answer is an object of F/[G,T]F / [G,T] not F/TF / T

  1. The internal hom constructed in that answer is in fact an object of /T\mathscr{F}/T. Recall that the phrase “the pullback of a map BCB \to C along a map ACA \to C” refers to the projection map A× CBAA\times_C B \to A.

    • CommentRowNumber9.
    • CommentAuthormaxsnew
    • CommentTimeJan 17th 2024

    Ah of course.

    It occurs to me that this monoidal structure is actually the same as the monoidal structure on a glued category in Section 4 of Hyland and Schalk by representing the monoid object in the topos as a lax monoidal functor 1H1 \to H where 11 is trivially monoidal closed.

    • CommentRowNumber10.
    • CommentAuthormaxsnew
    • CommentTimeJan 19th 2024

    Mention that Day convolution and slice over a monoid are always biclosed monoidal

    diff, v11, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2024

    I have hyperlinked that to doubly closed monoidal category.

    diff, v12, current