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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.
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.)
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
$\array{ &&&& \mathcal{C} \\ && & {}^{\mathllap{p}_1}\swarrow && \searrow^{\mathrlap{p_2}} \\ && \mathcal{A} && && \mathcal{B} \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} && {}^{\mathllap{i_3}}\swarrow && \searrow^{\mathrlap{i_4}} \\ \mathcal{X} && && \mathcal{Y} && && \mathcal{Z} }$Then we get functors
$\mathcal{X} \stackrel{F_1 \coloneqq (i_1)_!(i_2)^\ast}{\leftarrow} \mathcal{Y} \stackrel{F_2 \coloneqq (i_3)_!(i_4)^\ast}{\leftarrow} \mathcal{Z}$and the mate construction for the top square in the above diagram gives a composition transformation (at least if the BC condition holds)
$F_1 \circ F_2 \Rightarrow F_3$where
$F_3 \coloneqq (i_1 p_1)_! (i_4 p_2)^\ast \,.$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?
To check that I’m understanding: is it right that the canonical transformation is actually $F_3 \Rightarrow F_1 \circ F_2$, and the BC condition makes it invertible (and hence in particular can go the other direction)?
Yes.
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.
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?
$C=$ the interval category $(0\to 1)$, $D=C'=$ the terminal category, $\phi=0$, $\alpha=1$, so that $D'=\emptyset$, but $\alpha^*\phi_!$ is the identity functor.
Thanks! I have added that to the entry.
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?
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?
I guess I thought commutativity of substitution and dependent sum at least would show up somewhere.
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.
Ah yes, of course. Thanks.
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).
This is better:
added pointer also to
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]
added pointer also to:
finally for discussion in the generality of $\infty$-categories:
Urs Schreiber, around Def. 5.5 of: Quantization via Linear homotopy types (schreiber) [arXiv:1402.7041]
Dennis Gaitsgory, Jacob Lurie, Section 2.4.1 of: Weil’s conjecture for function fields (2014-2017) [“first volume of expanded account” pdf]
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).
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.
added pointer to:
Michael Shulman, e.g. Thm. 9.3 in: Enriched indexed categories, Theory Appl. Categ. 28 (2013) 616-695 [doi:1212.3914, tac:28-21]
Emily Riehl, Dominic Verity, Prop. 5.3.9 in: Kan extensions and the calculus of modules for ∞-categories, Algebr. Geom. Topol. 17 (2017) 189-271 [doi:10.2140/agt.2017.17.189, arXiv:1507.01460]
added pointer to:
and
added these pointers:
Michael Shulman, Comparing composites of left and right derived functors, New York Journal of Mathematics 17 (2011) 75-125 [arXiv:0706.2868, eudml:229181]
Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories, 20 18 (2008) 650–738 [tac:2018, arXiv:0706.1286]
I’d like the entry to state the following fact, if it is true:
For
$V$ some cosmos for enrichment,
$S \subset VCat$ some sub-1-category of small strict $V$-categories
$\mathbf{C}$ a (co)complete closed monoidal $V$-category
then the indexed monoidal $V$-category of $\mathbf{C}$-valued (co)presheaves on the $\mathcal{X}$s
$(\mathcal{X} \in S) \mapsto V Func( \mathcal{X} ,\, \mathbf{C} )$satisfies Beck-Chevalley.
In looking through the literature for who might have said this, I find Mike’s Theorem p. 9.3 here, whose proof seems like it claims a general statement of which this is a special case, if I am unwinding the notation correctly.(?)
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