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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.

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeJun 17th 2023

    I expanded on the observation that PSh(C)PSh(C) is the free cocompletion of CC by proving that PSh(C)PSh(C) is the local right adjoint to the free cocompletion functor.

    diff, v23, current

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeJun 17th 2023
    • (edited Jun 17th 2023)

    There are gaps in my argument (and in the resulting exposition) so I’ve reverted. But I’ve saved the contents here in case it can be salvaged.

    There _is_ a covariant functor with the same values on objects as $PSh$ that makes this a natural equivalence, namely the "free completion under small colimits" functor. This is, in fact, the [[adjoint (infinity,1)-functor#category_of_adjunctions | local left adjoint ]] to $PSh$, as can be observed by rewriting the equivalence in terms of right adjoints. If $D$ is a [[presentable (∞,1)-category]], let $Func^R(D, PSh(C)) \subseteq Func(D, PSh(C))$ be the full subcategory of accessible functors that preserve limits (i.e. functors that are right adjoints). Then +-- {: .un_lemma} ###### Lemma There is an [[equivalence of quasi-categories]], natural in small (∞,1)-categories $C$ and presentable (∞,1)-categories $D$ $$ Func^R(D, PSh(C))^{op} \to Func(C, D) $$ =-- +-- {: .proof} ###### Proof Taking an exponential transpose, we can identify $Func^R(D, PSh(C)) \subseteq \Func(D \times C^{op}, \infty Gpd)$ as the full subcategory of functors $F$ such that every $F(-, c)$ is accessible and preserves limits. Thus, we can compute natural equivalences $$ Func^R(D, PSh(C))^{op} \simeq Fun(C^{op}, Func^R(D, \infty Gpd))^{op} \simeq Fun(C, Func^L(\infty Gpd, D)) \simeq Fun(C, D) $$ =--
    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeJun 17th 2023

    I just realized that this page doesn’t discuss the functorality of PSh(C)PSh(C) in CC at all. I had mistakenly thought it took the contravariant functorality implied by Fun(,Gpd)Fun(-, \infty Gpd).

    So I’ve removed my remark about functorality, since it doesn’t make sense in context. Instead, a new section about functorality should probably be created. I’ll work on that tomorrow.

    diff, v24, current

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeJun 18th 2023

    I’ve written the promised section on functorality. I think I’ve filled the gaps in my proof that free cocompletion is the local left adjoint to Fun(,Grpd)Fun(-, \infty Grpd), by reducing it to the question of automorphisms of Cat\infty Cat.

    diff, v25, current

    • CommentRowNumber7.
    • CommentAuthorHurkyl
    • CommentTimeJun 18th 2023

    No, I’ve still made an error; I’ve shown that there is a natural equivalence between the free cocompletion and the local left adjoint to Fun(,Gpd)Fun(-, \infty Gpd), and for each SS the relationship between the yoneda embedding and insertion into the free cocompletion is determined by an automorphism of SS, but not that these automorphisms are natural in SS.

    There has to be a nice way to show these are homotopic, but I’m having a blind spot here.

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeJun 18th 2023

    Okay, I’ve finished off the proof. No more gaps, I hope.

    This exercise in lining up the details has been a lot more complicated than I thought it would be.

    diff, v26, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2023
    • (edited Jun 18th 2023)

    I have made the section “Functoriality” a subsection (now here) of the “Properties”-section, right after that on the free colimit completion.

    Where the (-) ladj{(\text{-})}^{ladj}- and (-) radj{(\text{-})}^{radj}-notation is used, I have added more explicit pointer to its definition, namely to this Lemma.

    diff, v27, current