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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2018
   • (edited Nov 26th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt 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](https://ncatlab.org/nlab/show/(infinity%2C1)-category+of+(infinity%2C1)-presheaves#SlicingCommutesWithFormingPresheaves)) 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? <a href="https://ncatlab.org/nlab/revision/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves/18">v18</a>, <a href="https://ncatlab.org/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves">current</a>

   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?

   diff, v18, current

2. • CommentRowNumber2.
   • CommentAuthorAli Caglayan
   • CommentTimeNov 26th 2018
   • (edited Nov 26th 2018)
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexHow 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.

   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.