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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2021
    • (edited Oct 12th 2021)

    am giving this its own entry, for ease of hyperlinking.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2021

    Just a note for later when the page can be edited again:

    A model-category version of the “fundamental theorem of \infty-topos theory” is also discussed in:

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    Finally coming back to this.

    I am looking for the generalization of the equivalence (of \infty-categories)

    PSh(𝒮) /y(X)Γ ()()PSh(𝒮 /X) PSh(\mathcal{S})_{/y(X)} \underoverset {\sim} { \Gamma_{(-)}(-) } {\longrightarrow} PSh \big( \mathcal{S}_{/X} \big)

    to the case that XX is not necessarily an object of 𝒮\mathcal{S} but is an object of PSh(𝒮)PSh(\mathcal{S}) (hence omitting the “y()y(-)” in the above formula), and interpreting 𝒮 /X\mathcal{S}_{/X} as the evident sub-category of PSh(𝒮) /XPSh(\mathcal{S})_{/X}.

    I vaguley thought it was proven in this generality in HTT, but now I don’t find it there, and maybe it isn’t. (Corollary 6.3.5.1 in HTT does generalize the above statement, but in another direction).

    In the special case that XPSh(𝒮)X \,\in\, PSh(\mathcal{S}) is 1-truncated, the statement is essentially Thm. 4.4 of Hollander’s article above (after identifying 𝒮 /X\mathcal{S}_{/X} with the Grothendieck construction that she refers to).

    For a while this was all I needed. But now I need the statement for 33-truncated XX, and it shouldn’t depend on the truncation of XX at all.

    Is there any reference?

    • CommentRowNumber4.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    The equivalence of these presheaf categories is corollary 5.1.6.12 of HTT; the last thing proved in the section on complete compactness. However, the equivalence constructed there is the colimit-preserving functor extending the map C /pP(C) /ypC_{/p} \to P(C)_{/yp}.

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    (I’m confused. I’m pretty sure I’ve made an error in the following, but I can’t find it. It’s probably just reversing an orientation so I’m leaving it up)

    For completeness, let p:JCp : J \to C be such that X=lim(yp)X = \lim(yp). C /pCC_{/p} \to C is a pullback of (C J) [1]C J×C J(C^J)^{[1]} \to C^J \times C^J, and so it is contravariant Grothendieck construction el¯ C(C J(Δ,p))\overline{el}_C( C^J(\Delta-, p)). However,

    C J(Δ,p)PSh(C) J(yΔ,yp)PSh(C) J(Δy,yp)PSh(C)(y,X)X C^J(\Delta-, p) \simeq PSh(C)^J(y \Delta -, yp) \simeq PSh(C)^J(\Delta y -, yp) \simeq PSh(C)(y-, X) \simeq X

    so C /pel¯ C(X)C_{/p} \simeq \overline{el}_C(X), and the isomorphism is PSh(el¯ C(X))PSh(C) /XPSh(\overline{el}_C(X)) \simeq PSh(C)_{/X}.

    • CommentRowNumber6.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    I guess the part that threw me off is that 5.1.6.12 isn’t actually full generality like I had remembered; I guess it only applies when XX is a limit of representable functors.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2022

    Thanks for looking into this! Would be a great issue to settle.

    Other people went through thinking that HTT 5.1.6.12 (right, that’s what I had meant to point to) gives the answer – e.g. MO:a/86209.

    Myself, I don’t see, either, how HTT 5.1.6.12 would give the answer (which is what I meant by saying that it “generalizes in a different direction”) – but I’d be happy to be educated if it does.

    Alternatively one could try to figure out if Hollander really needs her assumption that XX be 1-truncated. Possibly the only reason for this condition is that it allowed her to think of the slice in classical terms. But I haven’t dug through her proof yet.

    • CommentRowNumber8.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    Okay, here’s a way to establish the equivalence. The isomorphism PSh(C /c)PSh(C) /ycPSh(C_{/c}) \to PSh(C)_{/yc} when applied to an arrow f:abf : a \to b is the dependent sum PSh(C) /yaPSh(C) /ybPSh(C)_{/ya} \to PSh(C)_{/yb} and the left Kan extension PSh(C /a)PSh(C /b)PSh(C_{/a}) \to PSh(C_{/b}). (when viewing PShPSh as Fun(,Gpd)Fun(-, \infty Gpd))

    Taking local right adjoints, an arrow aba \to b acts on PSh(C) /ybPSh(C) /yaPSh(C)_{/yb} \to PSh(C)_{/ya} by taking pullbacks and on PSh(C /b)PSh(C /a)PSh(C_{/b}) \to PSh(C_{/a}) by the restriction map (along the composition map C /aC /bC_{/a} \to C_{/b}).

    Let p:JCp : J \to C be such that X=colim(yp)X = colim(yp). Then we have an isomorphism lim jPSh(C /p(j))lim jPSh(C) /yp(j)\lim_j PSh(C_{/p(j)}) \simeq \lim_j PSh(C)_{/yp(j)}. On the left side, we have

    lim jPSh(C /p(j))PSh(colim jC /p(j))PSh(el¯ C(X)) \lim_j PSh(C_{/p(j)}) \simeq PSh(\colim_j C_{/p(j)}) \simeq PSh(\overline{el}_C(X))

    where the latter isomorphism is given by computing the colimit in right fibrations over CC via the Grothendieck construction. On the right side, we have

    lim jPSh(C) /yp(j)PSh(C) /colim jyp(j)PSh(C) /X \lim_j PSh(C)_{/yp(j)} \simeq PSh(C)_{/\colim_j yp(j)} \simeq PSh(C)_{/X}

    by applying HTT proposition 6.3.5.14 which implies slicing gives a limit-preserving functor PSh(C) opLTopPSh(C)^{op} \to LTop and HTT 6.3.2.3 which says limits in LTopLTop are computed in Cat^\widehat{\infty Cat}.

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    I was hoping there was a simpler proof in terms of right fibrations; the proposition we’re asking for is that given a right fibration XCX \to C, we have

    RFib(C) /XRFib(X) RFib(C)_{/X} \simeq RFib(X)

    These are full subcategories of (,1)Cat /X(\infty,1)Cat_{/X} spanned by, respectively:

    • functors AXA \to X such that the composite AXCA \to X \to C is a right fibration
    • functors AXA \to X that are right fibrations

    It’s not true that these are the same subcategory is it? This would imply that if gg and gfgf are right fibrations, then ff is a right fibration, but I didn’t see anything like that flipping through HTT or the nLab pages.

    So, if these are isomorphic \infty-categories, the isomorphism RFib(X)RFib(C) /XRFib(X) \to RFib(C)_{/X} would have to be more complicated than just composing with XCX \to C.

    • CommentRowNumber10.
    • CommentAuthorHurkyl
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    Actually, I think that cancellation property is true. Characterizing right fibrations by the homotopy lifting property, given f:ABf : A \to B and g:BCg:B \to C, we have

    • Since gg is a right fibration, B [1]B× CC [1]B^{[1]} \simeq B \times_C C^{[1]}
    • Since gfg f is a right fibration, A [1]A× CC [1]A^{[1]} \simeq A \times_C C^{[1]}
    • Thus A [1]A× BB [1]A^{[1]} \simeq A \times_B B^{[1]}
    • Thus ff is a right fibration

    (for some reason, tikzcd wasn’t working so I couldn’t draw the diagram)

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 9th 2022
    • (edited Jul 9th 2022)

    This equivalence is very easy to implement as a Quillen equivalence of simplicial presheaves and their slice categories.

    First, the functor Γ:PSh(S)/X⟶PSh(S/X) is a right Quillen functor between projective model structures. This follows from the fact that fibrations and acyclic fibrations are stable under base changes in any model category.

    Next, the left adjoint functor PSh(S/X)→PSh(S)/X sends Čech nerves of open covers in S/X to Čech nerves of open covers in PSh(S). Thus, the Quillen adjunction descends to a Quillen adjunction between local projective model structures.

    Finally, we have to show that the derived unit and derived counit of this Quillen adjunction are weak equivalences.

    To this end we observe first that the right adjoint functor preserves weak equivalences and homotopy colimits. This implies that the derived unit and derived counit maps depend homotopy cocontinuously on the object. Thus, to show they are weak equivalences, it suffices to do it for the appropriate sets of (homotopy) generators.

    Given a cofibrant object APSh(S/X)A\in PSh(S/X), we can take it to be representable, in which case the unit map is an isomorphism.

    Given a fibrant object BPSh(S)/XB\in PSh(S)/X, we can take its underlying object in PSh(S)PSh(S) to be representable, in which case the counit map is an isomorphism.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2022

    Thanks for all this. I’ll think about it.

    Meanwhile, I was wondering about a general abstract argument. The following should be the easy proof in every context in which “category theory works”:

    Using

    (X,f X)(limis X(i),(f s X(i)) i)limi(s X(i),f s X(i)) \big( X, \, f_X \big) \;\simeq\; \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \, s_X(i) ,\, (f_{s_X(i)})_{i \in \mathcal{I}} \big) \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( s_X(i) ,\, f_{s_X(i)} \big)

    we check fully faithfulness by the following sequence of natural equivalences:

    PSh(𝒮) /B((X,f X),(X,f X)) PSh(𝒮)(X,X)×PSh(𝒮)(X,B){f X} limiPSh(𝒮)(s X(i),X)×limiPSh(𝒮)(s X(i),B){(f s X(i)) i} limi(PSh(𝒮)(s X(i),X)×PSh(𝒮)(s X(i),B){f s X(i)}) limi(limiPSh(𝒮)(s X(i),s X(i))×PSh(𝒮)(s X(i),B){f s X(i)}) limilimi(PSh(𝒮 /B)((s X(i),f s X(i)),(s X(i),f s X(i)))) PSh(𝒮 /B)(limi(s X(i),f s X(i)),limi(s X(i),f s X(i)))) PSh(𝒮 /B)((X,f x),(X,f X))) \begin{array}{l} PSh(\mathcal{S})_{/B} \Big( \big( X, \, f_X \big) \,, \big( X ,\, f_{X'} \big) \Big) \\ \;\simeq\; PSh(\mathcal{S}) \big( X \,, X' \big) \underset{ PSh(\mathcal{S}) \big( X \,, B \big) }{\times} \{f_X\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{S}) \big( s_X(i) \,, X' \big) \underset{ \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \Big\{ \big( f_{s_X(i)} \big)_{i \in \mathcal{I}} \Big\} \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( PSh(\mathcal{S}) \big( s_X(i) \,, X' \big) \underset{ PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \big\{ f_{s_X(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \bigg( \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} PSh(\mathcal{S}) \big( s_X(i) \,, s_{X'}(i') \big) \underset{ PSh(\mathcal{S}) \big( s_X(i) \,, B \big) }{\times} \big\{ f_{s_X(i)} \big\} \bigg) \\ \;\simeq\; \underset{ \underset{i \in \mathcal{I}}{\longleftarrow} }{\lim} \, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \, \bigg( PSh(\mathcal{S}_{/B}) \Big( \big( s_X(i) ,\, f_{s_X(i)} \big) \,, \big( s_{X'}(i') ,\, f_{s_{X'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{S}_{/B}) \Big( \underset{ \underset{i \in \mathcal{I}}{\longrightarrow} }{\lim} \big( s_X(i) ,\, f_{s_X(i)} \big) \,, \underset{ \underset{i' \in \mathcal{I}'}{\longrightarrow} }{\lim} \big( s_{X'}(i') ,\, f_{s_{X'}(i')} \big) \Big) \bigg) \\ \;\simeq\; PSh(\mathcal{S}_{/B}) \Big( \big( X ,\, f_x \big) \,, \big( X' ,\, f_{X'} \big) \Big) \bigg) \end{array}
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2022

    On the other hand, the previous proof over a representable which I had typed up In October 2021 (here at slice of presheaves is presheaves on slice) works verbatim also for non-representable base objects, and in this case gives the adjoint equivalence whose right adjoint is given by pullback.

    This is what you (Dmitri) are appealing to in the first statement of #11:

    First, the functor Γ:PSh(S)/X⟶PSh(S/X) is a right Quillen functor between projective model structures. This follows from the fact that fibrations and acyclic fibrations are stable under base changes in any model category.

    and that is the kind of argument that has been being spelled out in the above entry here. There I had found it necessary to add a few more observations to really exhibit a simplicial Quillen equivalence for enriched simplicial presheaves over an sSetsSet-site, but it’s all straightforward, of course.

    What I had not done before is type up any generalization of this argument to non-trivial Grothendieck topologies, which is of course the bulk of the argument in #11 above. Would be nice to type this up in the nnLab entry here on the “fundamental theorem”.

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 10th 2022

    I added the Čech-local argument to slice of presheaves is presheaves on slice. At some point the article switches from arbitrary presheaves to representable presheaves, for no particular reason, perhaps it should be generalized?

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