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nLab > Latest Changes: Beck-Chevalley condition

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2012

    at Beck-Chevalley condition I have added an Examples-section Pullbacks of opfibrations with statement and proof that the diagram of presheaves induced by a pullback of a small obfibration satisfies Beck-Chevalley.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2012

    Added to Beck-Chevalley condition a remark on its interpretation In logic / type theory.

    (This can be said more nicely and in more detail, but that’s what I have for the moment.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2013
    • (edited Dec 7th 2013)

    I have added the Beck-Chevalley-setup as an example to mate, just to have the cross-link.

    Here is a Question:

    Suppose we have a composite diagram of Wirthmüller contexts of the form

    𝒞 p 1↙ ↘ p 2 𝒜 ℬ i 1↙ ↘ i 2 i 3↙ ↘ i 4 𝒳 𝒴 𝒵 \array{ &&&& \mathcal{C} \\ && & {}^{\mathllap{p}_1}\swarrow && \searrow^{\mathrlap{p_2}} \\ && \mathcal{A} && && \mathcal{B} \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} && {}^{\mathllap{i_3}}\swarrow && \searrow^{\mathrlap{i_4}} \\ \mathcal{X} && && \mathcal{Y} && && \mathcal{Z} }

    Then we get functors

    𝒳←F 1≔(i 1) !(i 2) *𝒴←F 2≔(i 3) !(i 4) *𝒵 \mathcal{X} \stackrel{F_1 \coloneqq (i_1)_!(i_2)^\ast}{\leftarrow} \mathcal{Y} \stackrel{F_2 \coloneqq (i_3)_!(i_4)^\ast}{\leftarrow} \mathcal{Z}

    and the mate construction for the top square in the above diagram gives a composition transformation (at least if the BC condition holds)

    F 1∘F 2⇒F 3 F_1 \circ F_2 \Rightarrow F_3

    where

    F 3≔(i 1p 1) !(i 4p 2) *. F_3 \coloneqq (i_1 p_1)_! (i_4 p_2)^\ast \,.

    Further describing this kind of situation for longer sequences of spans of Wirthmüller contexts, which I will refrain from here in this comment, yields a “monad-oid” and its Kleisli composition, as discussed in another thread.

    Does this have an established name for the above kind of situation?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 8th 2013

    To check that I’m understanding: is it right that the canonical transformation is actually F 3⇒F 1∘F 2F_3 \Rightarrow F_1 \circ F_2, and the BC condition makes it invertible (and hence in particular can go the other direction)?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 8th 2013

    Yes.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2016

    I have added a Beck-Chevalley condition in the section Examples – For categories of presheaves a pointer to Joyal’s document on quasicategories, for the corresponding statement.

    What would be a quick example to show that the condition that the original square in this case be not just a pullback, but a pullback of opfibrations is a necessary condition?

    Then I did a little reorganization of the entry: There used to be a subsection “Bibrations” under the section “Examples”. This subsection had its own sub-sub-section “Examples”.

    So I fixed that: I moved the paragraphs on BC for bifibrations to the Definition-section (maybe some haronization is now in order there), and merged its list of Examples into the general list of examples.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeSep 1st 2016

    What would be a quick example to show that the condition that the original square in this case be not just a pullback, but a pullback of opfibrations is a necessary condition?

    C=C= the interval category (0→1)(0\to 1), D=C′=D=C'= the terminal category, ϕ=0\phi=0, α=1\alpha=1, so that D′=∅D'=\emptyset, but α *ϕ !\alpha^*\phi_! is the identity functor.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 1st 2016

    Thanks! I have added that to the entry.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2017

    I added Mackey’s restriction formula as an example.

    Presumably that’s something that could be readily shown in HoTT. Has there been any further work in HoTT concerning groups actions, etc.? Something seemed to have started a while ago, recall the regular action thread.

    By the way, why does Beck-Chevalley not appear in the HoTT book?

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeAug 6th 2017

    Presumably that’s something that could be readily shown in HoTT.

    Probably, although you’d first need to define some group theory, including colimits of diagrams of abelian groups.

    Has there been any further work in HoTT concerning groups actions, etc.?

    I don’t know of any.

    By the way, why does Beck-Chevalley not appear in the HoTT book?

    Where would it go?

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 7th 2017

    I guess I thought commutativity of substitution and dependent sum at least would show up somewhere.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeAug 7th 2017

    The nice thing about type theory is that commutativity of everything with substitution is built into the notation, so you don’t even need to mention it. Of course it has to be mentioned when constructing categorical semantics, but the HoTT book is always working internally to type theory.

    • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 7th 2017

    Ah yes, of course. Thanks.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    The list of references in Beck-Chevalley condition is unsatisfactory. Is there some textbook or similar which reviews/discusses the issue in some generality for a general audience?

    Searching, I have found

    which I have added now to the list. But this is not what I am looking for (it’s too specialized).

    diff, v28, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    This is better:

    diff, v28, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    added pointer also to

    • Duško Pavlović, Section 1 of: Categorical interpolation: Descent and the Beck-Chevalley condition without direct images, in: Category Theory, Lecture Notes in Mathematics 1488 (1991) [doi:10.1007/BFb0084229, pdf]

    together with this quote:

    The Beck-Chevalley condition has arisen in the theory of descent - as developed from Grothendieck 1959. Jon Beck and Claude Chevalley studied it independently from each another. […] It is conspicuous that neither of them ever published anything on it. [Pavlović 1991, §14]

    diff, v28, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022

    added pointer also to:

    diff, v29, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2022
    • (edited Aug 7th 2022)

    finally for discussion in the generality of ∞\infty-categories:

    diff, v30, current

    • CommentRowNumber19.
    • CommentAuthorvarkor
    • CommentTimeSep 22nd 2022

    Added reference to exact square. More generally, it might be preferable to give this condition a more descriptive name, especially when the contribution of Beck to the definition of this condition is disputed (for instance, see the discussion on the categories mailing list in November 2007).

    diff, v32, current

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 22nd 2023

    Transferred the content of an answer by Fosco Loregian.

    diff, v35, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2023

    Thanks for the alert that this motivation was missing on the page. I feel like we may have this elsewhere on the nLab, but would have to check (not right now though, am just on my phone in between other thing). There is some more along these lines in “Quantization via Linear Homotopy Types”, but may not expository enough for what we’d want here.

    In any case, I have rewritten the text you dropped, for better clarity. The simple point is that Beck-Chevalley is the condition to make pull-push base change through correspondences functorial under composition of correspondences by fiber product of adjacent legs.

    I would also like to redo the diagrams (pity to use xymatrix and then still show composition from top-right to bottom-left instead of just left to right as usual) but maybe later.

    diff, v36, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2023

    okay, I have now fixed the orientation of the diagrams (here)

    and added also brief mentioning of the other main motivation for Beck-Chevalley as expressing substitution of free variables commuting with quantification over bound variables (here)

    diff, v36, current

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