nForum - Discussion Feed ((infinity,1)-category of (infinity,1)-presheaves) 2023-03-27T08:20:32+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Ali Caglayan comments on "(infinity,1)-category of (infinity,1)-presheaves" (74120) https://nforum.ncatlab.org/discussion/9311/?Focus=74120#Comment_74120 2018-11-26T09:27:57+00:00 2023-03-27T08:20:32+00:00 Ali Caglayan https://nforum.ncatlab.org/account/1731/ How would you describe PSh &infin;(&Cscr;)\mathrm{PSh}_\infty(\mathcal{C}) from inside &Cscr;\mathcal{C}? I don’t get how this is supposed to be done in HoTT. Edit: Nevermind I ...

How would you describe $\mathrm{PSh}_\infty(\mathcal{C})$ from inside $\mathcal{C}$? 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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Urs comments on "(infinity,1)-category of (infinity,1)-presheaves" (74119) https://nforum.ncatlab.org/discussion/9311/?Focus=74119#Comment_74119 2018-11-26T08:47:26+00:00 2023-03-27T08:20:32+00:00 Urs https://nforum.ncatlab.org/account/4/ It would be nice if the entry were a little more explicit about the slicing theorem PSh &infin;(&Cscr; /p)&rightarrow;&simeq;PSh &infin;(&Cscr;) /yp. ...

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

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

(here)

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

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

This must be an easy theorem in HoTT?

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