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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2018
    • (edited Nov 26th 2018)

    It would be nice if the entry were a little more explicit about the slicing theorem

    PSh (𝒞 /p)PSh (𝒞) /yp. PSh_\infty(\mathcal{C}_{/p}) \stackrel{\simeq}{\to} PSh_\infty(\mathcal{C})_{/y p} \,.


    In the special case that the small \infty-category 𝒞\mathcal{C} happens to be a small \infty-groupoid and that pp is constant on an object X𝒞X \in \mathcal{C}, it ought to be true that an explicit form of this equivalence is given in semi-HoTT notation by

    ((cγX)(γ))(ccγX(γ) cγX*). \left( (c \overset{\gamma}{\to}X) \;\mapsto\; \mathcal{F}(\gamma) \right) \;\mapsto\; \left( c \;\mapsto\; \array{ \underset{c \underset{\gamma}{\to}X}{\sum} \mathcal{F}(\gamma) \\ \downarrow \\ \underset{c \underset{\gamma}{\to}X}{\sum} \ast } \right) \,.

    This must be an easy theorem in HoTT?

    diff, v18, current

    • CommentRowNumber2.
    • CommentAuthorAli Caglayan
    • CommentTimeNov 26th 2018
    • (edited Nov 26th 2018)

    How would you describe PSh (𝒞)\mathrm{PSh}_\infty(\mathcal{C}) from inside 𝒞\mathcal{C}? I don’t get how this is supposed to be done in HoTT.

    Edit: Nevermind I haven’t read this properly.

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